Research ArticleOn the 119901-Version of the Schwab-Borchardt Mean
Edward Neuman
Mathematical Research Institute 144 Hawthorn Hollow Carbondale IL 62903 USA
Correspondence should be addressed to Edward Neuman edneuman76gmailcom
Received 10 January 2014 Accepted 28 April 2014 Published 13 May 2014
Academic Editor Kenneth S Berenhaut
Copyright copy 2014 Edward Neuman This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper deals with a one-parameter generalization of the Schwab-Borchardt mean The new mean is defined in terms ofthe inverse functions of the generalized trigonometric and generalized hyperbolic functions The four new bivariate means areintroduced as particular cases of the 119901-version of the Schwab-Borchardt mean For the particular value of the parameter 119901 thesemeans become either the classical logarithmic mean or the Seiffert means or the Neuman-Sandor mean Wilker- and Huygens-type inequalities involving inverse functions of the generalized trigonometric and the generalized hyperbolic functions are alsoestablished
1 Introduction
The Schwab-Borchardt mean of two numbers 119909 ge 0 and 119910 gt
0 denoted by SB(119909 119910) equiv SB is defined as
SB (119909 119910) =
radic1199102 minus 1199092
cosminus1 (119909119910) 0 le 119909 le 119910
radic1199092 minus 1199102
coshminus1 (119909119910) 119910 lt 119909
119909 119909 = 119910
(1)
(see [1 Thrm 84] [2 (23)]) It follows from (1) that SB(119909 119910)is not symmetric in its arguments and is a homogeneousfunction of degree 1 in 119909 and 119910 This mean has been studiedextensively in [1ndash5]
The goal of this paper is to define and investigate ageneralization of the Schwab-Borchardt mean SB The newone has a form which is similar to (1) but depends on aparameter 119901 which is used in definitions of two familiesof higher transcendental functions called the generalizedtrigonometric and the generalized hyperbolic functions Def-initions of these functions are given in Section 2 Also wewill use a notion of the 119877-hypergeometric functions of twovariablesTheir definition and somebasic properties are givenin Section 3 Throughout the sequel the 119901-V119890119903119904119894119900119899 of SB willbe denoted by SB
119901 The latter is introduced in Section 4
Therein somebasic properties of the newmean are discussedIn Section 5 we define four new bivariate means whichcan be considered as the generalized logarithmic mean thegeneralized Seiffert means and the generalized Neuman-Sandor mean In the last section of this paper we shallestablish inequalities involving new means as well as Wilker-and Huygens-type inequalities involving inverse functions ofthe generalized trigonometric and the generalized hyperbolicfunctions
2 Definitions of Generalized Trigonometricand Hyperbolic Functions
For the readerrsquos convenience we recall first definition of thecelebrated Gauss hypergeometric function 119865(119886 119887 119888 119911)
119865 (120572 120573 120574 119911) =
infin
sum
119899=0
(120572 119899) (120573 119899)
(120574 119899)
119911119899
119899 |119911| lt 1 (2)
where (120572 119899) = 120572(120572 + 1) sdot sdot sdot (120572 + 119899 minus 1) (119899 = 0) is the shiftedfactorial or Appell symbol with (120572 0) = 1 if 120572 = 0 and120574 = 0 minus1 minus2
In what follows we will always assume that the number119901 is strictly greater than 1 We will adopt notation anddefinitions used in [6] Let
120587119901= 2
120587119901
sin (120587119901) (3)
Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014 Article ID 697643 7 pageshttpdxdoiorg1011552014697643
2 International Journal of Mathematics and Mathematical Sciences
Further let
119886119901=120587119901
2 119887
119901= 2minus1119901
119865(1
1199011
119901 1 +
1
1199011
2)
119888119901= 2minus1119901
119865(11
119901 1 +
1
1199011
2)
(4)
Also let 119868 = (0 1) and let 119869 = (1infin) The generalizedtrigonometric and hyperbolic functions needed in this paperare the following homeomorphisms
sin119901 (0 119886
119901) 997888rarr 119868 cos
119901 (0 119886
119901) 997888rarr 119868
tan119901 (0 119887119901) 997888rarr 119868 sinh
119901 (0 119888119901) 997888rarr 119868
(5)
cosh119901 (0infin) 997888rarr 119869 tanh
119901 (0infin) 997888rarr 119869 (6)
The inverse functions of sinminus1119901
and sinhminus1119901
can be representedas follows [7]
sinminus1119901119906 = int
119906
0
(1 minus 119905119901)minus1119901
119889119905 = 119906119865(1
1199011
119901 1 +
1
119901 119906119901) (7)
sinhminus1119901119906 = int
119906
0
(1 + 119905119901)minus1119901
119889119905 = 119906119865(1
1199011
119901 1 +
1
119901 minus119906119901)
(8)
Inverse functions of the remaining four functions can beexpressed in terms of sinminus1
119901and sinhminus1
119901 We have
cosminus1119901119906 = sinminus1
119901(119901radic1 minus 119906119901) (9)
coshminus1119901119906 = sinhminus1
119901(119901radic119906119901 minus 1) (10)
tanminus1119901119906 = sinminus1
119901(
119906
119901radic1 + 119906119901)
tanhminus1119901119906 = sinhminus1
119901(
119906
119901radic1 minus 119906119901)
(11)
The generalized trigonometric functions have been intro-duced by Lindqvist in [8] It is obvious that the func-tions under discussion become classical trigonometric andhyperbolic functions when 119901 = 2 It is known thatthey are eigenfunctions of the Dirichlet problem for theone-dimensional 119901-Laplacian For more details concerninggeneralized trigonometric functions generalized hyperbolicfunctions and inequalities involving these functions theinterested reader is referred to [6ndash15]
3 The 119877-HypergeometricFunctions of Two Variables
In this section we give the definition of the bivariate 119877-hypergeometric functions which are used in the sequel Someresults for these functions are also included here
In what follows the symbols R+and R
gtwill stand for
the nonnegative semiaxis and the set of positive numbersrespectively Let 119887 = (119887
1 1198872) isin R2gt By 120583
119887 where
120583119887(119905) =
Γ (1198871+ 1198872)
Γ (1198871) Γ (1198872)1199051198871minus1(1 minus 119905)
1198872minus1 (12)
we will denote the Dirichlet measure on the interval [0 1] Itis well known that120583
119887is the probabilitymeasure on its domain
Also let 119883 = (119909 119910) isin R2gt In [16 17] the 119877-
hypergeometric function 119877120572(119887 119883) (120572 isin R) is defined as
where 119906 = (119905 1 minus 119905) and 119906 sdot 119883 = 119905119909 + (1 minus 119905)119910 arethe dot product of 119906 and 119883 Many of the important specialfunctions including Gaussrsquo hypergeometric function 119865 andsome elliptic integrals admit the integral representation (13)For more details the interested reader is referred to Carlsonrsquosmonograph [17]
A nice feature of the 119877-hypergeometric function is itspermutation symmetry in both parameters and variables thatis
(120574 gt 0)For the later use let us also recordCarlsonrsquos inequality [18
Theorem 3]
[119877120572(119887 119883)]
1120572
le [119877120573(119887 119883)]
1120573 (16)
(120572 120573 = 0 120572 le 120573)Wewill also need the following result which appears in [5
Proposition 21] Let 120572 lt 0 119887 isin R2+ and let 119883119884 isin R2
gt Then
the following inequality
119877120572(119887 120582119883 + (1 minus 120582) 119884) le [119877
120572(119887 119883)]
120582
[119877120572(119887 119884)]
1minus120582 (17)
holds true for all 0 le 120582 le 1
4 Definition and Basic Properties of the 119901-Version of SB
Let the numbers 119909 and 119910 have the meaning as in Section 1For the sake of presentation we recall first a formula for themean SB in terms of the 119877-hypergeometric function
SB (119909 119910) equiv SB = 119877minus12
(1
2 1 1199092 1199102)
minus1
(18)
(see [17 19])Wedefine the119901-version (119901 gt 1) of themean SB as follows
SB119901(119909 119910) equiv SB
119901= 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus1
(19)
The rightmost member of (19) is a special case of whatis called in mathematical literature the 119877-hypergeometricmean (see [2 17 18]) Using elementary properties in the 119877-hypergeometric functions we see that SB
119901(119909 119910) is the mean
International Journal of Mathematics and Mathematical Sciences 3
value of 119909 and 119910 Moreover this mean is nonsymmetric andhomogeneous of degree 1 in its variables The well-knownresults on the119877-hypergeometricmeans lead to the conclusionthat SB
119901is a strongly increasing function of the parameter 119901
For the brevity of notation let us introduce a particular119877-hypergeometric function
119877119878(119909 119910) = 119877
minus1119901(1
119901 1 119909 119910) (20)
Clearly function 119877119878is nonsymmetric and homogeneous of
degree minus1119901 in its variables Comparison with (19) yields
SB119901(119909 119910) = 119877
119878(119909119901 119910119901)minus1
(21)
We shall demonstrate now that SB119901can also be expressed
in terms of cosminus1119901
and coshminus1119901
SB119901(119909 119910) =
119901radic119910119901 minus 119909119901
cosminus1119901(119909119910)
0 le 119909 le 119910
119901radic119909119901 minus 119910119901
coshminus1119901(119909119910)
119910 lt 119909
119909 119909 = 119910
(22)
For the proof of the first part of (22) let us record aformulawhich shows that theGauss hypergeometric function119865 can be expressed in terms of the bivariate119877-hypergeometricfunction
(120573 120574 minus 120573 1 minus 119911 1) (23)
(see eg [17 ((59)ndash(12))]) Application of the last formula to(7) yields
sinminus1119901119906 = 119906119877
119878(1 minus 119906
119901 1) (24)
where 0 lt 119906 lt 1 This in conjunction with (9) gives
cosminus1119901119906 =
119901radic1 minus 119906119901119877119878(119906119901 1) (25)
Letting above 119906 = 119909119910 and utilizing homogeneity of thefunction 119877
119878 we obtain
cosminus1119901(119909
119910) =
119901radic119910119901 minus 119909119901119877119878(119909119901 119910119901)
=119901radic119910119901 minus 119909119901[SB
119901(119909 119910)]
minus1
(26)
where in the last stepwe have utilized formula (19)This yieldsthe first part of (22) The second part can be established inan analogous manner A key formula needed here reads asfollows
coshminus1119901119906 =
119901radic119906119901 minus 1119877119878(119906119901 1) (27)
119906 gt 1 We omit further detailsFunction 119877
119878admits an integral representation
119877119878(119909 119910) =
1
119901int
infin
0
(119905 + 119909)minus1119901
(119905 + 119910)minus1
119889119905 (28)
This follows from the known result [20 (19169)]
119877minus120572
(1205731 1205732 119909 119910) =
1
119861 (120572 1205721015840)int
infin
0
1199051205721015840minus1(119905 + 119909)
minus1205731(119905 + 119910)minus1205732
119889119905
(29)
where 119861 stands for the beta function and 1205721015840= 1205731+ 1205732minus 120572
Letting 120572 = 1205731= 1119901 and 120573
2= 1 we obtain 120572
1015840= 1 Formula
(28) now follows because 119861(1119901 1) = 119901
5 Four New Bivariate Means Derived from SB119901
The goal of this section is to define and investigate four newbivariate meansThey are defined in terms of the SB
119901and the
bivariate power mean 119860119903(119903 isin R) Recall that
119860119903(119909 119910) equiv 119860
119903=
119903radic119909119903+ 119910119903
2119903 = 0
radic119909119910 119903 = 0
(30)
where 119909 119910 gt 0 The power mean 1198600of order 0 is usually
denoted by 119866 and is called the geometric mean It is wellknown that the power mean 119860
119903is a strictly increasing
function of 119903We are in a position to define four new means of positive
numbers 119909 and 119910 In what follows these means will bedenoted by 119871
119901 119875119901119872119901and 119879
119901 where 119901 gt 1 They are defined
as follows
119871119901(119909 119910) equiv119871
119901= SB119901(1198601199012
119866) (31)
119875119901(119909 119910) equiv119875
119901= SB119901(119866 119860
1199012) (32)
119879119901(119909 119910) equiv119879
119901= SB119901(1198601199012
119860119901) (33)
119872119901(119909 119910) equiv119872
119901= SB119901(119860119901 1198601199012
) (34)
In the case when 119901 = 2 these means become the classicallogarithmicmean 119871 two Seiffert means119875 and119879 (see [21 22])and the Neuman-Sandor mean119872 introduced in [4]
For the later use we introduce quantity V119901 where
V119901= (
100381610038161003816100381610038161199091199012
minus 119910119901210038161003816100381610038161003816
1199091199012 + 1199101199012)
2119901
(35)
Clearly 0 lt V119901lt 1
The main result of this section reads as follows
Theorem 1 (let 119909 119910 gt 0) Then the following formulas
119871119901= 1198601199012
V119901
tanhminus1119901V119901
(36)
119875119901= 1198601199012
V119901
sinminus1119901V119901
(37)
119879119901= 1198601199012
V119901
tanminus1119901V119901
(38)
119872119901= 1198601199012
V119901
sinhminus1119901V119901
(39)
are valid
4 International Journal of Mathematics and Mathematical Sciences
Proof We begin with the proof of (36) Making use of (31)and (22) we obtain
119871119901=
119901radic119860119901
1199012minus 119866119901
coshminus1119901(1198601199012
119866) (40)
Elementary computations yield
119860119901
1199012minus 119866119901= (
1199091199012
+ 1199101199012
2)
2
minus (119909119910)1199012
= (1199091199012
minus 1199101199012
2)
2
(41)
Multiplying and dividing by
(1199091199012
+ 1199101199012
2)
2
(42)
we obtain using (35)
119860119901
1199012minus 119866119901= (1198601199012
V119901)119901
(43)
We shall write the denominator of (40) using (10) and (43) asfollows
coshminus1119901(1198601199012
119866) = tanhminus1
119901
119901radic1 minus (119866
1198601199012
)
119901
= tanhminus1119901
119901radic119860119901
1199012minus 119866119901
119860119901
1199012
= tanhminus1119901V119901
(44)
This in conjunction with (40) and (43) gives the desiredresult (36)
We shall provide now a sketch of the proof of formula(39) It follows from (22) and (34) that
119872119901=
119901radic119860119901
119901 minus 119860119901
1199012
coshminus1119901(1198601199011198601199012
) (45)
Elementary computations yield
119860119901
119901minus 119860119901
1199012= (1198601199012
V119901)119901
(46)
Application of (10) with 119906 = 1198601199011198601199012
givescoshminus1119901(1198601199011198601199012
) = sinhminus1119901V119901 This in conjunction with
(45) and (46) yields the asserted result (39) The remainingtwo formulas for the 119901-versions 119875
119901and 119879
119901of the Seiffert
means can be established in an analogous manner using (32)or (33) (22) and (9) We leave it to the interested reader Theproof is complete
6 Inequalities Involving the SB119901
Means
This section deals with inequalities involving the SB119901means
In particular inequalities for the four means introduced inSection 5 are established Also we shall prove Wilker-typeand Huygens-type inequalities involving inverse functions ofthe generalized trigonometric functions and the generalizedhyperbolic functions
Our first result reads as follows
Theorem 2 Let the positive numbers 119909 and 119910 be such that 119909 gt
119910 Then
119878119861119901(119909 119910) lt 119878119861
119901(119910 119909) (47)
Proof We shall prove the assertion using integral formula(28) and formula (21) Let 119906 gt 1 and let 119905 gt 0 Then 119906
119901gt 1
and
(119905 + 119906119901)1minus1119901
gt (119905 + 1)1minus1119901 (48)
or what is the same that
(119905 + 119906119901)minus1119901
(119905 + 1)minus1
gt (119905 + 119906119901)minus1
(119905 + 1)minus1119901 (49)
because 1 minus 1119901 gt 0 Integration yields
1
119901int
infin
0
(119905 + 119906119901)minus1119901
(119905 + 1)minus1119889119905
gt1
119901int
infin
0
(119905 + 119906119901)minus1
(119905 + 1)minus1119901
119889119905
(50)
or what is the same that the inequality
119877119878(119906119901 1) gt 119877
119878(1 119906119901) (51)
Raising both sides to the power of minus1 and next applyingformula (21) we obtain
SB119901(119906 1) lt SB
119901(1 119906) (52)
Letting 119906 = 119909119910 and next utilizing homogeneity of SB119901 we
obtain the desired result
The four new means defined in Section 5 and the powermeans are comparable We have the following
Proof Thefirst third fourth and the sixth inequalities in (53)follow from their definitions (see (31) (32) (34) and (33))The second and the fifth inequalities can be obtained usingTheorem 2 applied to twopairs of defining equations (31)-(32)and (34)-(33)
Our next result reads as follows
Theorem 4 Let 120572 = 1119901 and 120573 = 1minus1119901 Then the inequality
Applying the permutation symmetry (see (14)) to the R-hypergeometric function on the left-hand side and nextraising both sides of the resulting formula to the power of minus1we obtain
119877minus1119901
(1
119901 1 119910119901 119909119901)
minus1
119910119901minus1
= 119877minus1(1
119901 1 119909119901 119910119901)
minus1
(57)
Since minus1 lt minus1119901 Carlsonrsquos inequality (16) yields
119877minus1(1
119901 1 119909119901 119910119901)
minus1
lt 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus119901
(58)
Combining this with (57) we obtain
119877minus1119901
(1
119901 1 119910119901 119909119901)
minus1
119910119901minus1
lt 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus119901
(59)
Utilizing (20) and (21) we can write the last inequality in theform
SB119901(119910 119909) 119910
119901minus1lt SB119901(119909 119910)
119901
(60)
The desired inequality (54) now follows
Corollary 5 Let the numbers 120572 and 120573 be the same as inTheorem 4 Then the following inequalities
119875120572
119901119866120573lt 119871119901 (61)
119871120572
119901119860120573
1199012lt 119875119901 (62)
119879120572
119901119860120573
1199012lt 119872119901
119872120572
119901119860120573
119901lt 119879119901
(63)
hold true
Proof Inequality (61) can be obtained using Theorem 4 with119909 = 119860
1199012and 119910 = 119866 Next we utilize formulas (31) and (32)
to obtain the desired result In a similar fashion one can proveinequality (62) using Theorem 4 with 119909 = 119866 and 119910 = 119860
1199012
Making use of formulas (31) and (32) yields the assertionTheremaining inequalities (63) can be proven in an analogousmanner We omit further details
Another inequality for the SB119901mean is contained in the
following
Theorem 6 Let 1199091 1199092 1199101 1199102gt 0 Then
next we use (20) and finally we raise both sides of theresulting inequality to the power of minus2 This gives
119877119878(119909119901
1 119910119901
1)minus1
119877119878(119909119901
2 119910119901
2)minus1
le [119877119878(119860119901
119901 (1199091 1199092) 119860119901
119901 (1199101 1199102))minus1
]2
(66)
Application of (21) gives the desired inequality (64)
Corollary 7 Assume that the positive numbers 119909 and 119910 arenot equal Then the following inequalities
119875119901119872119901lt 1198602
1199012 (67)
119871119901119879119901lt 1198602
1199012(68)
hold true
Proof For the proof of (67) we let in (64) 1199091= 119866 119909
2= 119860119901
and 1199101= 1199102= 1198601199012
Then the left-hand side of (67) followsfrom (32) and (34) To obtain the right-hand side of theinequality in question let us notice that 119860
119901(119866 119860119901) = 119860
1199012
and obviously 119860119901(1198601199012
1198601199012
) = 1198601199012
Inequality (68) can beestablished in a similar manner We use (64) with 119909
1= 1199092=
1198601199012
1199101= 119866 and 119910
2= 119860119901followed by application of (31)
and (33)
We will close this section proving Wilker-type andHuygens-type inequalities which involve inverse functions ofthe generalized trigonometric and hyperbolic functions Tothis aim we shall employ the following result [23]
Theorem A Let 119906 V 120582 120583 be positive numbers Assume that 119906and V satisfy the separation condition
119906 lt 1 lt V (69)
Then the inequality
1 lt120582
120582 + 120583119906119903+
120583
120582 + 120583V119904 (70)
holds true if either
1 lt 119906120574V120575 119904 gt 0 119903120582 le
119904120583120574
120575(71)
6 International Journal of Mathematics and Mathematical Sciences
or
119906120574V120575 lt 1 119904 lt 0 119903120582 le
119904120583120574
120575 (72)
for some 120574 120575 ge 0with 120574+120575 = 1 If 119906 and V satisfy the separationcondition (69) together with
1 lt 1205741
119906+ 120575
1
V (73)
then inequality (70) is also valid if
119903 le 119904 le minus1 120583120574 le 120582120575 (74)
As in the previous sections the letters 119909 and 119910 will standfor positive and unequal numbers Also 119866119860
1199012and 119860
119901
denote the power means of 119909 and 119910For the sake of notation we define
120591 = 119901119905 where 119905 =1
2log(119909
119910) (75)
We are in a position to prove the following
Theorem 8 Let V119901be the same as in (35) and let 120590
1=
(cosh2(1205912))1119901 Assume that 120582 120583 119904 gt 0 Also let
(119906 V) = (sinminus1119901V119901
tanhminus1119901V119901
1205901
V119901
tanhminus1119901V119901
) (76)
or
(119906 V) = (tanhminus1119901V119901
sinhminus1119901V119901
V119901
sinhminus1119901V119901
) (77)
If
(119901 minus 1) 119903120582 le 119904120583 (78)
then inequality (70) is satisfied if (119906 V) is defined either in (76)or in (77) Inequality (70) is also satisfied if either
(119906 V) = (V119901
sinminus1119901V119901
tanhminus1119901V119901
sinminus1119901V119901
) (79)
or
(119906 V) = (1205902
V119901
tanminus1119901V119901
sinhminus1119901V119901
tanminus1119901V119901
) (80)
provided that
119903120582 le (119901 minus 1) 119904120583 (81)
Here 1205902= 1205901(cosh 120591)1119901
Proof For the proof of validity of (70) with (119906 V) as definedin (76) we let
119906 =119871119901
119875119901
V =119871119901
119866 (82)
It follows from (53) that the separation condition (69) issatisfied To complete the proof of (76) we utilize a well-known fact about bivariatemeans Let119873(119909 119910) equiv 119873 be ameanwhich is homogeneous of degree 1 in its variables Then
This togetherwith (36) and (37) gives the explicit formula (76)for (119906 V) We will show now that the first inequality in (71) issatisfied if 120574 = 1119901 and 120575 = 1 minus 1119901 To this aim we utilize(82) and write inequality (61) as follows
1 lt 119906120572V120573 (85)
where 120572 = 1119901 and 120573 = 1 minus 1119901 This yields 120574 = 120572
and 120575 = 120573 To obtain condition (78) of validity of (70) wesubstitute 120574120575 = 1(119901minus 1) into the last inequality in (71)Thiscompletes the first part of the proof Assume now that (119906 V)is the same as is defined in (79) We will prove that (70) holdstrue provided that condition (81) is satisfied First we define
119906 =119875119901
1198601199012
V =119875119901
119871119901
(86)
Again we appeal to (53) to claim that 119906 and V satisfy theseparation condition (69) Making use of (36) and (37) weobtain an explicit formula (79) for (119906 V)Wewill show that thefirst inequality in (71) is satisfied if 120574 = 1minus1119901 and 120575 = 1119901 Tothis aim we utilize (86) and write inequality (62) as follows
1 lt 119906120573V120572 (87)
where 120572 = 1119901 and 120573 = 1 minus 1119901 To prove that (70) holdstrue if (81) is satisfied we substitute 120574120575 = 119901 minus 1 into the lastinequality in (71) The assertion now follows The remainingtwo cases when (119906 V) is defined in (77) or in (80) can beestablished in the analogous manner In these cases we haveeither
(119906 V) = (119872119901
119879119901
119872119901
1198601199012
) (88)
or
(119906 V) = (119879119901
119860119901
119879119901
119872119901
) (89)
We leave it to the reader to complete the proof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] J M Borwein and P B Borwein Pi and the AGM A Study inAnalytic Number Theory and Computational Complexity JohnWiley amp Sons New York NY USA 1987
[2] B C Carlson ldquoAlgorithms involving arithmetic and geometricmeansrdquoThe American Mathematical Monthly vol 78 pp 496ndash505 1971
[3] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[4] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[5] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[6] B A Bhayo and M Vuorinen ldquoInequalities for eigenfunctionsof the 119901-Laplacianrdquo Issues of Analysis vol 2 no 1 pp 13ndash352013
[7] R Klen M Vuorinen and X Zhang ldquoInequalities for the gen-eralized trigonometric and hyperbolic functionsrdquo Journal ofMathematical Analysis and Applications vol 409 no 1 pp 521ndash529 2014
[8] P Lindqvist ldquoSome remarkable sine and cosine functionsrdquo Ric-erche di Matematica vol 44 no 2 pp 269ndash290 1995
[9] A Baricz B A Bhayo and R Klen ldquoConvexity properties ofgeneralized trigonometric and hyperbolic functionsrdquo Aequa-tiones Mathematicae 2013
[10] B A Bhayo ldquoPowermean inequality of generalized trigonomet-ric functionsrdquo 2012 httparxivorgabs12090873
[11] B A Bhayo and M Vuorinen ldquoOn generalized trigonometricfunctions with two parametersrdquo Journal of ApproximationTheory vol 164 no 10 pp 1415ndash1426 2012
[12] W D Jiang M KWang Y M Chu Y P Jiang and F Qi ldquoCon-vexity of the generalized sine function and the generalized hyp-erbolic sine functionrdquo Journal of ApproximationTheory vol 174pp 1ndash9 2013
[13] E Neuman ldquoInequalities involving inverse circular and inversehyperbolic functionsrdquo Journal of Mathematical Inequalities vol4 no 1 pp 11ndash14 2010
[14] S Takeuchi ldquoGeneralized Jacobian elliptic functions and theirapplication to bifurcation problems associated with 119901-Lapla-cianrdquo Journal of Mathematical Analysis and Applications vol385 no 1 pp 24ndash35 2012
[15] S Takeuchi ldquoThe basis property of generalized Jacobian ellipticfunctionsrdquo 2013 httparxivorgabs13100597
[16] B C Carlson ldquoSome inequalities for hypergeometric func-tionsrdquo Proceedings of the American Mathematical Society vol17 pp 32ndash39 1966
[17] B C Carlson Special Functions of AppliedMathematics Acade-mic Press New York NY USA 1977
[18] B CCarlson ldquoAhypergeometricmean valuerdquoProceedings of theAmerican Mathematical Society vol 16 pp 759ndash766 1965
[19] J L Brenner and B C Carlson ldquoHomogeneous mean valuesweights and asymptoticsrdquo Journal of Mathematical Analysis andApplications vol 123 no 1 pp 265ndash280 1987
[20] F W J Olver D W Lozier R F Boisvert and C W ClarkTheNISTHandbook ofMathematical Functions CambridgeUniver-sity Press New York NY USA 2010
[21] H J Seiffert ldquoProblem 887rdquo Nieuw Archief voor Wiskunde vol11 p 176 1993
[22] H J Seiffert ldquoAufgabe 16rdquoWurzel vol 29 p 87 1995[23] E Neuman ldquoInequalities for weighted sums of powers and their
2 International Journal of Mathematics and Mathematical Sciences
Further let
119886119901=120587119901
2 119887
119901= 2minus1119901
119865(1
1199011
119901 1 +
1
1199011
2)
119888119901= 2minus1119901
119865(11
119901 1 +
1
1199011
2)
(4)
Also let 119868 = (0 1) and let 119869 = (1infin) The generalizedtrigonometric and hyperbolic functions needed in this paperare the following homeomorphisms
sin119901 (0 119886
119901) 997888rarr 119868 cos
119901 (0 119886
119901) 997888rarr 119868
tan119901 (0 119887119901) 997888rarr 119868 sinh
119901 (0 119888119901) 997888rarr 119868
(5)
cosh119901 (0infin) 997888rarr 119869 tanh
119901 (0infin) 997888rarr 119869 (6)
The inverse functions of sinminus1119901
and sinhminus1119901
can be representedas follows [7]
sinminus1119901119906 = int
119906
0
(1 minus 119905119901)minus1119901
119889119905 = 119906119865(1
1199011
119901 1 +
1
119901 119906119901) (7)
sinhminus1119901119906 = int
119906
0
(1 + 119905119901)minus1119901
119889119905 = 119906119865(1
1199011
119901 1 +
1
119901 minus119906119901)
(8)
Inverse functions of the remaining four functions can beexpressed in terms of sinminus1
119901and sinhminus1
119901 We have
cosminus1119901119906 = sinminus1
119901(119901radic1 minus 119906119901) (9)
coshminus1119901119906 = sinhminus1
119901(119901radic119906119901 minus 1) (10)
tanminus1119901119906 = sinminus1
119901(
119906
119901radic1 + 119906119901)
tanhminus1119901119906 = sinhminus1
119901(
119906
119901radic1 minus 119906119901)
(11)
The generalized trigonometric functions have been intro-duced by Lindqvist in [8] It is obvious that the func-tions under discussion become classical trigonometric andhyperbolic functions when 119901 = 2 It is known thatthey are eigenfunctions of the Dirichlet problem for theone-dimensional 119901-Laplacian For more details concerninggeneralized trigonometric functions generalized hyperbolicfunctions and inequalities involving these functions theinterested reader is referred to [6ndash15]
3 The 119877-HypergeometricFunctions of Two Variables
In this section we give the definition of the bivariate 119877-hypergeometric functions which are used in the sequel Someresults for these functions are also included here
In what follows the symbols R+and R
gtwill stand for
the nonnegative semiaxis and the set of positive numbersrespectively Let 119887 = (119887
1 1198872) isin R2gt By 120583
119887 where
120583119887(119905) =
Γ (1198871+ 1198872)
Γ (1198871) Γ (1198872)1199051198871minus1(1 minus 119905)
1198872minus1 (12)
we will denote the Dirichlet measure on the interval [0 1] Itis well known that120583
119887is the probabilitymeasure on its domain
Also let 119883 = (119909 119910) isin R2gt In [16 17] the 119877-
hypergeometric function 119877120572(119887 119883) (120572 isin R) is defined as
where 119906 = (119905 1 minus 119905) and 119906 sdot 119883 = 119905119909 + (1 minus 119905)119910 arethe dot product of 119906 and 119883 Many of the important specialfunctions including Gaussrsquo hypergeometric function 119865 andsome elliptic integrals admit the integral representation (13)For more details the interested reader is referred to Carlsonrsquosmonograph [17]
A nice feature of the 119877-hypergeometric function is itspermutation symmetry in both parameters and variables thatis
(120574 gt 0)For the later use let us also recordCarlsonrsquos inequality [18
Theorem 3]
[119877120572(119887 119883)]
1120572
le [119877120573(119887 119883)]
1120573 (16)
(120572 120573 = 0 120572 le 120573)Wewill also need the following result which appears in [5
Proposition 21] Let 120572 lt 0 119887 isin R2+ and let 119883119884 isin R2
gt Then
the following inequality
119877120572(119887 120582119883 + (1 minus 120582) 119884) le [119877
120572(119887 119883)]
120582
[119877120572(119887 119884)]
1minus120582 (17)
holds true for all 0 le 120582 le 1
4 Definition and Basic Properties of the 119901-Version of SB
Let the numbers 119909 and 119910 have the meaning as in Section 1For the sake of presentation we recall first a formula for themean SB in terms of the 119877-hypergeometric function
SB (119909 119910) equiv SB = 119877minus12
(1
2 1 1199092 1199102)
minus1
(18)
(see [17 19])Wedefine the119901-version (119901 gt 1) of themean SB as follows
SB119901(119909 119910) equiv SB
119901= 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus1
(19)
The rightmost member of (19) is a special case of whatis called in mathematical literature the 119877-hypergeometricmean (see [2 17 18]) Using elementary properties in the 119877-hypergeometric functions we see that SB
119901(119909 119910) is the mean
International Journal of Mathematics and Mathematical Sciences 3
value of 119909 and 119910 Moreover this mean is nonsymmetric andhomogeneous of degree 1 in its variables The well-knownresults on the119877-hypergeometricmeans lead to the conclusionthat SB
119901is a strongly increasing function of the parameter 119901
For the brevity of notation let us introduce a particular119877-hypergeometric function
119877119878(119909 119910) = 119877
minus1119901(1
119901 1 119909 119910) (20)
Clearly function 119877119878is nonsymmetric and homogeneous of
degree minus1119901 in its variables Comparison with (19) yields
SB119901(119909 119910) = 119877
119878(119909119901 119910119901)minus1
(21)
We shall demonstrate now that SB119901can also be expressed
in terms of cosminus1119901
and coshminus1119901
SB119901(119909 119910) =
119901radic119910119901 minus 119909119901
cosminus1119901(119909119910)
0 le 119909 le 119910
119901radic119909119901 minus 119910119901
coshminus1119901(119909119910)
119910 lt 119909
119909 119909 = 119910
(22)
For the proof of the first part of (22) let us record aformulawhich shows that theGauss hypergeometric function119865 can be expressed in terms of the bivariate119877-hypergeometricfunction
(120573 120574 minus 120573 1 minus 119911 1) (23)
(see eg [17 ((59)ndash(12))]) Application of the last formula to(7) yields
sinminus1119901119906 = 119906119877
119878(1 minus 119906
119901 1) (24)
where 0 lt 119906 lt 1 This in conjunction with (9) gives
cosminus1119901119906 =
119901radic1 minus 119906119901119877119878(119906119901 1) (25)
Letting above 119906 = 119909119910 and utilizing homogeneity of thefunction 119877
119878 we obtain
cosminus1119901(119909
119910) =
119901radic119910119901 minus 119909119901119877119878(119909119901 119910119901)
=119901radic119910119901 minus 119909119901[SB
119901(119909 119910)]
minus1
(26)
where in the last stepwe have utilized formula (19)This yieldsthe first part of (22) The second part can be established inan analogous manner A key formula needed here reads asfollows
coshminus1119901119906 =
119901radic119906119901 minus 1119877119878(119906119901 1) (27)
119906 gt 1 We omit further detailsFunction 119877
119878admits an integral representation
119877119878(119909 119910) =
1
119901int
infin
0
(119905 + 119909)minus1119901
(119905 + 119910)minus1
119889119905 (28)
This follows from the known result [20 (19169)]
119877minus120572
(1205731 1205732 119909 119910) =
1
119861 (120572 1205721015840)int
infin
0
1199051205721015840minus1(119905 + 119909)
minus1205731(119905 + 119910)minus1205732
119889119905
(29)
where 119861 stands for the beta function and 1205721015840= 1205731+ 1205732minus 120572
Letting 120572 = 1205731= 1119901 and 120573
2= 1 we obtain 120572
1015840= 1 Formula
(28) now follows because 119861(1119901 1) = 119901
5 Four New Bivariate Means Derived from SB119901
The goal of this section is to define and investigate four newbivariate meansThey are defined in terms of the SB
119901and the
bivariate power mean 119860119903(119903 isin R) Recall that
119860119903(119909 119910) equiv 119860
119903=
119903radic119909119903+ 119910119903
2119903 = 0
radic119909119910 119903 = 0
(30)
where 119909 119910 gt 0 The power mean 1198600of order 0 is usually
denoted by 119866 and is called the geometric mean It is wellknown that the power mean 119860
119903is a strictly increasing
function of 119903We are in a position to define four new means of positive
numbers 119909 and 119910 In what follows these means will bedenoted by 119871
119901 119875119901119872119901and 119879
119901 where 119901 gt 1 They are defined
as follows
119871119901(119909 119910) equiv119871
119901= SB119901(1198601199012
119866) (31)
119875119901(119909 119910) equiv119875
119901= SB119901(119866 119860
1199012) (32)
119879119901(119909 119910) equiv119879
119901= SB119901(1198601199012
119860119901) (33)
119872119901(119909 119910) equiv119872
119901= SB119901(119860119901 1198601199012
) (34)
In the case when 119901 = 2 these means become the classicallogarithmicmean 119871 two Seiffert means119875 and119879 (see [21 22])and the Neuman-Sandor mean119872 introduced in [4]
For the later use we introduce quantity V119901 where
V119901= (
100381610038161003816100381610038161199091199012
minus 119910119901210038161003816100381610038161003816
1199091199012 + 1199101199012)
2119901
(35)
Clearly 0 lt V119901lt 1
The main result of this section reads as follows
Theorem 1 (let 119909 119910 gt 0) Then the following formulas
119871119901= 1198601199012
V119901
tanhminus1119901V119901
(36)
119875119901= 1198601199012
V119901
sinminus1119901V119901
(37)
119879119901= 1198601199012
V119901
tanminus1119901V119901
(38)
119872119901= 1198601199012
V119901
sinhminus1119901V119901
(39)
are valid
4 International Journal of Mathematics and Mathematical Sciences
Proof We begin with the proof of (36) Making use of (31)and (22) we obtain
119871119901=
119901radic119860119901
1199012minus 119866119901
coshminus1119901(1198601199012
119866) (40)
Elementary computations yield
119860119901
1199012minus 119866119901= (
1199091199012
+ 1199101199012
2)
2
minus (119909119910)1199012
= (1199091199012
minus 1199101199012
2)
2
(41)
Multiplying and dividing by
(1199091199012
+ 1199101199012
2)
2
(42)
we obtain using (35)
119860119901
1199012minus 119866119901= (1198601199012
V119901)119901
(43)
We shall write the denominator of (40) using (10) and (43) asfollows
coshminus1119901(1198601199012
119866) = tanhminus1
119901
119901radic1 minus (119866
1198601199012
)
119901
= tanhminus1119901
119901radic119860119901
1199012minus 119866119901
119860119901
1199012
= tanhminus1119901V119901
(44)
This in conjunction with (40) and (43) gives the desiredresult (36)
We shall provide now a sketch of the proof of formula(39) It follows from (22) and (34) that
119872119901=
119901radic119860119901
119901 minus 119860119901
1199012
coshminus1119901(1198601199011198601199012
) (45)
Elementary computations yield
119860119901
119901minus 119860119901
1199012= (1198601199012
V119901)119901
(46)
Application of (10) with 119906 = 1198601199011198601199012
givescoshminus1119901(1198601199011198601199012
) = sinhminus1119901V119901 This in conjunction with
(45) and (46) yields the asserted result (39) The remainingtwo formulas for the 119901-versions 119875
119901and 119879
119901of the Seiffert
means can be established in an analogous manner using (32)or (33) (22) and (9) We leave it to the interested reader Theproof is complete
6 Inequalities Involving the SB119901
Means
This section deals with inequalities involving the SB119901means
In particular inequalities for the four means introduced inSection 5 are established Also we shall prove Wilker-typeand Huygens-type inequalities involving inverse functions ofthe generalized trigonometric functions and the generalizedhyperbolic functions
Our first result reads as follows
Theorem 2 Let the positive numbers 119909 and 119910 be such that 119909 gt
119910 Then
119878119861119901(119909 119910) lt 119878119861
119901(119910 119909) (47)
Proof We shall prove the assertion using integral formula(28) and formula (21) Let 119906 gt 1 and let 119905 gt 0 Then 119906
119901gt 1
and
(119905 + 119906119901)1minus1119901
gt (119905 + 1)1minus1119901 (48)
or what is the same that
(119905 + 119906119901)minus1119901
(119905 + 1)minus1
gt (119905 + 119906119901)minus1
(119905 + 1)minus1119901 (49)
because 1 minus 1119901 gt 0 Integration yields
1
119901int
infin
0
(119905 + 119906119901)minus1119901
(119905 + 1)minus1119889119905
gt1
119901int
infin
0
(119905 + 119906119901)minus1
(119905 + 1)minus1119901
119889119905
(50)
or what is the same that the inequality
119877119878(119906119901 1) gt 119877
119878(1 119906119901) (51)
Raising both sides to the power of minus1 and next applyingformula (21) we obtain
SB119901(119906 1) lt SB
119901(1 119906) (52)
Letting 119906 = 119909119910 and next utilizing homogeneity of SB119901 we
obtain the desired result
The four new means defined in Section 5 and the powermeans are comparable We have the following
Proof Thefirst third fourth and the sixth inequalities in (53)follow from their definitions (see (31) (32) (34) and (33))The second and the fifth inequalities can be obtained usingTheorem 2 applied to twopairs of defining equations (31)-(32)and (34)-(33)
Our next result reads as follows
Theorem 4 Let 120572 = 1119901 and 120573 = 1minus1119901 Then the inequality
Applying the permutation symmetry (see (14)) to the R-hypergeometric function on the left-hand side and nextraising both sides of the resulting formula to the power of minus1we obtain
119877minus1119901
(1
119901 1 119910119901 119909119901)
minus1
119910119901minus1
= 119877minus1(1
119901 1 119909119901 119910119901)
minus1
(57)
Since minus1 lt minus1119901 Carlsonrsquos inequality (16) yields
119877minus1(1
119901 1 119909119901 119910119901)
minus1
lt 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus119901
(58)
Combining this with (57) we obtain
119877minus1119901
(1
119901 1 119910119901 119909119901)
minus1
119910119901minus1
lt 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus119901
(59)
Utilizing (20) and (21) we can write the last inequality in theform
SB119901(119910 119909) 119910
119901minus1lt SB119901(119909 119910)
119901
(60)
The desired inequality (54) now follows
Corollary 5 Let the numbers 120572 and 120573 be the same as inTheorem 4 Then the following inequalities
119875120572
119901119866120573lt 119871119901 (61)
119871120572
119901119860120573
1199012lt 119875119901 (62)
119879120572
119901119860120573
1199012lt 119872119901
119872120572
119901119860120573
119901lt 119879119901
(63)
hold true
Proof Inequality (61) can be obtained using Theorem 4 with119909 = 119860
1199012and 119910 = 119866 Next we utilize formulas (31) and (32)
to obtain the desired result In a similar fashion one can proveinequality (62) using Theorem 4 with 119909 = 119866 and 119910 = 119860
1199012
Making use of formulas (31) and (32) yields the assertionTheremaining inequalities (63) can be proven in an analogousmanner We omit further details
Another inequality for the SB119901mean is contained in the
following
Theorem 6 Let 1199091 1199092 1199101 1199102gt 0 Then
next we use (20) and finally we raise both sides of theresulting inequality to the power of minus2 This gives
119877119878(119909119901
1 119910119901
1)minus1
119877119878(119909119901
2 119910119901
2)minus1
le [119877119878(119860119901
119901 (1199091 1199092) 119860119901
119901 (1199101 1199102))minus1
]2
(66)
Application of (21) gives the desired inequality (64)
Corollary 7 Assume that the positive numbers 119909 and 119910 arenot equal Then the following inequalities
119875119901119872119901lt 1198602
1199012 (67)
119871119901119879119901lt 1198602
1199012(68)
hold true
Proof For the proof of (67) we let in (64) 1199091= 119866 119909
2= 119860119901
and 1199101= 1199102= 1198601199012
Then the left-hand side of (67) followsfrom (32) and (34) To obtain the right-hand side of theinequality in question let us notice that 119860
119901(119866 119860119901) = 119860
1199012
and obviously 119860119901(1198601199012
1198601199012
) = 1198601199012
Inequality (68) can beestablished in a similar manner We use (64) with 119909
1= 1199092=
1198601199012
1199101= 119866 and 119910
2= 119860119901followed by application of (31)
and (33)
We will close this section proving Wilker-type andHuygens-type inequalities which involve inverse functions ofthe generalized trigonometric and hyperbolic functions Tothis aim we shall employ the following result [23]
Theorem A Let 119906 V 120582 120583 be positive numbers Assume that 119906and V satisfy the separation condition
119906 lt 1 lt V (69)
Then the inequality
1 lt120582
120582 + 120583119906119903+
120583
120582 + 120583V119904 (70)
holds true if either
1 lt 119906120574V120575 119904 gt 0 119903120582 le
119904120583120574
120575(71)
6 International Journal of Mathematics and Mathematical Sciences
or
119906120574V120575 lt 1 119904 lt 0 119903120582 le
119904120583120574
120575 (72)
for some 120574 120575 ge 0with 120574+120575 = 1 If 119906 and V satisfy the separationcondition (69) together with
1 lt 1205741
119906+ 120575
1
V (73)
then inequality (70) is also valid if
119903 le 119904 le minus1 120583120574 le 120582120575 (74)
As in the previous sections the letters 119909 and 119910 will standfor positive and unequal numbers Also 119866119860
1199012and 119860
119901
denote the power means of 119909 and 119910For the sake of notation we define
120591 = 119901119905 where 119905 =1
2log(119909
119910) (75)
We are in a position to prove the following
Theorem 8 Let V119901be the same as in (35) and let 120590
1=
(cosh2(1205912))1119901 Assume that 120582 120583 119904 gt 0 Also let
(119906 V) = (sinminus1119901V119901
tanhminus1119901V119901
1205901
V119901
tanhminus1119901V119901
) (76)
or
(119906 V) = (tanhminus1119901V119901
sinhminus1119901V119901
V119901
sinhminus1119901V119901
) (77)
If
(119901 minus 1) 119903120582 le 119904120583 (78)
then inequality (70) is satisfied if (119906 V) is defined either in (76)or in (77) Inequality (70) is also satisfied if either
(119906 V) = (V119901
sinminus1119901V119901
tanhminus1119901V119901
sinminus1119901V119901
) (79)
or
(119906 V) = (1205902
V119901
tanminus1119901V119901
sinhminus1119901V119901
tanminus1119901V119901
) (80)
provided that
119903120582 le (119901 minus 1) 119904120583 (81)
Here 1205902= 1205901(cosh 120591)1119901
Proof For the proof of validity of (70) with (119906 V) as definedin (76) we let
119906 =119871119901
119875119901
V =119871119901
119866 (82)
It follows from (53) that the separation condition (69) issatisfied To complete the proof of (76) we utilize a well-known fact about bivariatemeans Let119873(119909 119910) equiv 119873 be ameanwhich is homogeneous of degree 1 in its variables Then
This togetherwith (36) and (37) gives the explicit formula (76)for (119906 V) We will show now that the first inequality in (71) issatisfied if 120574 = 1119901 and 120575 = 1 minus 1119901 To this aim we utilize(82) and write inequality (61) as follows
1 lt 119906120572V120573 (85)
where 120572 = 1119901 and 120573 = 1 minus 1119901 This yields 120574 = 120572
and 120575 = 120573 To obtain condition (78) of validity of (70) wesubstitute 120574120575 = 1(119901minus 1) into the last inequality in (71)Thiscompletes the first part of the proof Assume now that (119906 V)is the same as is defined in (79) We will prove that (70) holdstrue provided that condition (81) is satisfied First we define
119906 =119875119901
1198601199012
V =119875119901
119871119901
(86)
Again we appeal to (53) to claim that 119906 and V satisfy theseparation condition (69) Making use of (36) and (37) weobtain an explicit formula (79) for (119906 V)Wewill show that thefirst inequality in (71) is satisfied if 120574 = 1minus1119901 and 120575 = 1119901 Tothis aim we utilize (86) and write inequality (62) as follows
1 lt 119906120573V120572 (87)
where 120572 = 1119901 and 120573 = 1 minus 1119901 To prove that (70) holdstrue if (81) is satisfied we substitute 120574120575 = 119901 minus 1 into the lastinequality in (71) The assertion now follows The remainingtwo cases when (119906 V) is defined in (77) or in (80) can beestablished in the analogous manner In these cases we haveeither
(119906 V) = (119872119901
119879119901
119872119901
1198601199012
) (88)
or
(119906 V) = (119879119901
119860119901
119879119901
119872119901
) (89)
We leave it to the reader to complete the proof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] J M Borwein and P B Borwein Pi and the AGM A Study inAnalytic Number Theory and Computational Complexity JohnWiley amp Sons New York NY USA 1987
[2] B C Carlson ldquoAlgorithms involving arithmetic and geometricmeansrdquoThe American Mathematical Monthly vol 78 pp 496ndash505 1971
[3] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[4] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[5] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[6] B A Bhayo and M Vuorinen ldquoInequalities for eigenfunctionsof the 119901-Laplacianrdquo Issues of Analysis vol 2 no 1 pp 13ndash352013
[7] R Klen M Vuorinen and X Zhang ldquoInequalities for the gen-eralized trigonometric and hyperbolic functionsrdquo Journal ofMathematical Analysis and Applications vol 409 no 1 pp 521ndash529 2014
[8] P Lindqvist ldquoSome remarkable sine and cosine functionsrdquo Ric-erche di Matematica vol 44 no 2 pp 269ndash290 1995
[9] A Baricz B A Bhayo and R Klen ldquoConvexity properties ofgeneralized trigonometric and hyperbolic functionsrdquo Aequa-tiones Mathematicae 2013
[10] B A Bhayo ldquoPowermean inequality of generalized trigonomet-ric functionsrdquo 2012 httparxivorgabs12090873
[11] B A Bhayo and M Vuorinen ldquoOn generalized trigonometricfunctions with two parametersrdquo Journal of ApproximationTheory vol 164 no 10 pp 1415ndash1426 2012
[12] W D Jiang M KWang Y M Chu Y P Jiang and F Qi ldquoCon-vexity of the generalized sine function and the generalized hyp-erbolic sine functionrdquo Journal of ApproximationTheory vol 174pp 1ndash9 2013
[13] E Neuman ldquoInequalities involving inverse circular and inversehyperbolic functionsrdquo Journal of Mathematical Inequalities vol4 no 1 pp 11ndash14 2010
[14] S Takeuchi ldquoGeneralized Jacobian elliptic functions and theirapplication to bifurcation problems associated with 119901-Lapla-cianrdquo Journal of Mathematical Analysis and Applications vol385 no 1 pp 24ndash35 2012
[15] S Takeuchi ldquoThe basis property of generalized Jacobian ellipticfunctionsrdquo 2013 httparxivorgabs13100597
[16] B C Carlson ldquoSome inequalities for hypergeometric func-tionsrdquo Proceedings of the American Mathematical Society vol17 pp 32ndash39 1966
[17] B C Carlson Special Functions of AppliedMathematics Acade-mic Press New York NY USA 1977
[18] B CCarlson ldquoAhypergeometricmean valuerdquoProceedings of theAmerican Mathematical Society vol 16 pp 759ndash766 1965
[19] J L Brenner and B C Carlson ldquoHomogeneous mean valuesweights and asymptoticsrdquo Journal of Mathematical Analysis andApplications vol 123 no 1 pp 265ndash280 1987
[20] F W J Olver D W Lozier R F Boisvert and C W ClarkTheNISTHandbook ofMathematical Functions CambridgeUniver-sity Press New York NY USA 2010
[21] H J Seiffert ldquoProblem 887rdquo Nieuw Archief voor Wiskunde vol11 p 176 1993
[22] H J Seiffert ldquoAufgabe 16rdquoWurzel vol 29 p 87 1995[23] E Neuman ldquoInequalities for weighted sums of powers and their
International Journal of Mathematics and Mathematical Sciences 3
value of 119909 and 119910 Moreover this mean is nonsymmetric andhomogeneous of degree 1 in its variables The well-knownresults on the119877-hypergeometricmeans lead to the conclusionthat SB
119901is a strongly increasing function of the parameter 119901
For the brevity of notation let us introduce a particular119877-hypergeometric function
119877119878(119909 119910) = 119877
minus1119901(1
119901 1 119909 119910) (20)
Clearly function 119877119878is nonsymmetric and homogeneous of
degree minus1119901 in its variables Comparison with (19) yields
SB119901(119909 119910) = 119877
119878(119909119901 119910119901)minus1
(21)
We shall demonstrate now that SB119901can also be expressed
in terms of cosminus1119901
and coshminus1119901
SB119901(119909 119910) =
119901radic119910119901 minus 119909119901
cosminus1119901(119909119910)
0 le 119909 le 119910
119901radic119909119901 minus 119910119901
coshminus1119901(119909119910)
119910 lt 119909
119909 119909 = 119910
(22)
For the proof of the first part of (22) let us record aformulawhich shows that theGauss hypergeometric function119865 can be expressed in terms of the bivariate119877-hypergeometricfunction
(120573 120574 minus 120573 1 minus 119911 1) (23)
(see eg [17 ((59)ndash(12))]) Application of the last formula to(7) yields
sinminus1119901119906 = 119906119877
119878(1 minus 119906
119901 1) (24)
where 0 lt 119906 lt 1 This in conjunction with (9) gives
cosminus1119901119906 =
119901radic1 minus 119906119901119877119878(119906119901 1) (25)
Letting above 119906 = 119909119910 and utilizing homogeneity of thefunction 119877
119878 we obtain
cosminus1119901(119909
119910) =
119901radic119910119901 minus 119909119901119877119878(119909119901 119910119901)
=119901radic119910119901 minus 119909119901[SB
119901(119909 119910)]
minus1
(26)
where in the last stepwe have utilized formula (19)This yieldsthe first part of (22) The second part can be established inan analogous manner A key formula needed here reads asfollows
coshminus1119901119906 =
119901radic119906119901 minus 1119877119878(119906119901 1) (27)
119906 gt 1 We omit further detailsFunction 119877
119878admits an integral representation
119877119878(119909 119910) =
1
119901int
infin
0
(119905 + 119909)minus1119901
(119905 + 119910)minus1
119889119905 (28)
This follows from the known result [20 (19169)]
119877minus120572
(1205731 1205732 119909 119910) =
1
119861 (120572 1205721015840)int
infin
0
1199051205721015840minus1(119905 + 119909)
minus1205731(119905 + 119910)minus1205732
119889119905
(29)
where 119861 stands for the beta function and 1205721015840= 1205731+ 1205732minus 120572
Letting 120572 = 1205731= 1119901 and 120573
2= 1 we obtain 120572
1015840= 1 Formula
(28) now follows because 119861(1119901 1) = 119901
5 Four New Bivariate Means Derived from SB119901
The goal of this section is to define and investigate four newbivariate meansThey are defined in terms of the SB
119901and the
bivariate power mean 119860119903(119903 isin R) Recall that
119860119903(119909 119910) equiv 119860
119903=
119903radic119909119903+ 119910119903
2119903 = 0
radic119909119910 119903 = 0
(30)
where 119909 119910 gt 0 The power mean 1198600of order 0 is usually
denoted by 119866 and is called the geometric mean It is wellknown that the power mean 119860
119903is a strictly increasing
function of 119903We are in a position to define four new means of positive
numbers 119909 and 119910 In what follows these means will bedenoted by 119871
119901 119875119901119872119901and 119879
119901 where 119901 gt 1 They are defined
as follows
119871119901(119909 119910) equiv119871
119901= SB119901(1198601199012
119866) (31)
119875119901(119909 119910) equiv119875
119901= SB119901(119866 119860
1199012) (32)
119879119901(119909 119910) equiv119879
119901= SB119901(1198601199012
119860119901) (33)
119872119901(119909 119910) equiv119872
119901= SB119901(119860119901 1198601199012
) (34)
In the case when 119901 = 2 these means become the classicallogarithmicmean 119871 two Seiffert means119875 and119879 (see [21 22])and the Neuman-Sandor mean119872 introduced in [4]
For the later use we introduce quantity V119901 where
V119901= (
100381610038161003816100381610038161199091199012
minus 119910119901210038161003816100381610038161003816
1199091199012 + 1199101199012)
2119901
(35)
Clearly 0 lt V119901lt 1
The main result of this section reads as follows
Theorem 1 (let 119909 119910 gt 0) Then the following formulas
119871119901= 1198601199012
V119901
tanhminus1119901V119901
(36)
119875119901= 1198601199012
V119901
sinminus1119901V119901
(37)
119879119901= 1198601199012
V119901
tanminus1119901V119901
(38)
119872119901= 1198601199012
V119901
sinhminus1119901V119901
(39)
are valid
4 International Journal of Mathematics and Mathematical Sciences
Proof We begin with the proof of (36) Making use of (31)and (22) we obtain
119871119901=
119901radic119860119901
1199012minus 119866119901
coshminus1119901(1198601199012
119866) (40)
Elementary computations yield
119860119901
1199012minus 119866119901= (
1199091199012
+ 1199101199012
2)
2
minus (119909119910)1199012
= (1199091199012
minus 1199101199012
2)
2
(41)
Multiplying and dividing by
(1199091199012
+ 1199101199012
2)
2
(42)
we obtain using (35)
119860119901
1199012minus 119866119901= (1198601199012
V119901)119901
(43)
We shall write the denominator of (40) using (10) and (43) asfollows
coshminus1119901(1198601199012
119866) = tanhminus1
119901
119901radic1 minus (119866
1198601199012
)
119901
= tanhminus1119901
119901radic119860119901
1199012minus 119866119901
119860119901
1199012
= tanhminus1119901V119901
(44)
This in conjunction with (40) and (43) gives the desiredresult (36)
We shall provide now a sketch of the proof of formula(39) It follows from (22) and (34) that
119872119901=
119901radic119860119901
119901 minus 119860119901
1199012
coshminus1119901(1198601199011198601199012
) (45)
Elementary computations yield
119860119901
119901minus 119860119901
1199012= (1198601199012
V119901)119901
(46)
Application of (10) with 119906 = 1198601199011198601199012
givescoshminus1119901(1198601199011198601199012
) = sinhminus1119901V119901 This in conjunction with
(45) and (46) yields the asserted result (39) The remainingtwo formulas for the 119901-versions 119875
119901and 119879
119901of the Seiffert
means can be established in an analogous manner using (32)or (33) (22) and (9) We leave it to the interested reader Theproof is complete
6 Inequalities Involving the SB119901
Means
This section deals with inequalities involving the SB119901means
In particular inequalities for the four means introduced inSection 5 are established Also we shall prove Wilker-typeand Huygens-type inequalities involving inverse functions ofthe generalized trigonometric functions and the generalizedhyperbolic functions
Our first result reads as follows
Theorem 2 Let the positive numbers 119909 and 119910 be such that 119909 gt
119910 Then
119878119861119901(119909 119910) lt 119878119861
119901(119910 119909) (47)
Proof We shall prove the assertion using integral formula(28) and formula (21) Let 119906 gt 1 and let 119905 gt 0 Then 119906
119901gt 1
and
(119905 + 119906119901)1minus1119901
gt (119905 + 1)1minus1119901 (48)
or what is the same that
(119905 + 119906119901)minus1119901
(119905 + 1)minus1
gt (119905 + 119906119901)minus1
(119905 + 1)minus1119901 (49)
because 1 minus 1119901 gt 0 Integration yields
1
119901int
infin
0
(119905 + 119906119901)minus1119901
(119905 + 1)minus1119889119905
gt1
119901int
infin
0
(119905 + 119906119901)minus1
(119905 + 1)minus1119901
119889119905
(50)
or what is the same that the inequality
119877119878(119906119901 1) gt 119877
119878(1 119906119901) (51)
Raising both sides to the power of minus1 and next applyingformula (21) we obtain
SB119901(119906 1) lt SB
119901(1 119906) (52)
Letting 119906 = 119909119910 and next utilizing homogeneity of SB119901 we
obtain the desired result
The four new means defined in Section 5 and the powermeans are comparable We have the following
Proof Thefirst third fourth and the sixth inequalities in (53)follow from their definitions (see (31) (32) (34) and (33))The second and the fifth inequalities can be obtained usingTheorem 2 applied to twopairs of defining equations (31)-(32)and (34)-(33)
Our next result reads as follows
Theorem 4 Let 120572 = 1119901 and 120573 = 1minus1119901 Then the inequality
Applying the permutation symmetry (see (14)) to the R-hypergeometric function on the left-hand side and nextraising both sides of the resulting formula to the power of minus1we obtain
119877minus1119901
(1
119901 1 119910119901 119909119901)
minus1
119910119901minus1
= 119877minus1(1
119901 1 119909119901 119910119901)
minus1
(57)
Since minus1 lt minus1119901 Carlsonrsquos inequality (16) yields
119877minus1(1
119901 1 119909119901 119910119901)
minus1
lt 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus119901
(58)
Combining this with (57) we obtain
119877minus1119901
(1
119901 1 119910119901 119909119901)
minus1
119910119901minus1
lt 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus119901
(59)
Utilizing (20) and (21) we can write the last inequality in theform
SB119901(119910 119909) 119910
119901minus1lt SB119901(119909 119910)
119901
(60)
The desired inequality (54) now follows
Corollary 5 Let the numbers 120572 and 120573 be the same as inTheorem 4 Then the following inequalities
119875120572
119901119866120573lt 119871119901 (61)
119871120572
119901119860120573
1199012lt 119875119901 (62)
119879120572
119901119860120573
1199012lt 119872119901
119872120572
119901119860120573
119901lt 119879119901
(63)
hold true
Proof Inequality (61) can be obtained using Theorem 4 with119909 = 119860
1199012and 119910 = 119866 Next we utilize formulas (31) and (32)
to obtain the desired result In a similar fashion one can proveinequality (62) using Theorem 4 with 119909 = 119866 and 119910 = 119860
1199012
Making use of formulas (31) and (32) yields the assertionTheremaining inequalities (63) can be proven in an analogousmanner We omit further details
Another inequality for the SB119901mean is contained in the
following
Theorem 6 Let 1199091 1199092 1199101 1199102gt 0 Then
next we use (20) and finally we raise both sides of theresulting inequality to the power of minus2 This gives
119877119878(119909119901
1 119910119901
1)minus1
119877119878(119909119901
2 119910119901
2)minus1
le [119877119878(119860119901
119901 (1199091 1199092) 119860119901
119901 (1199101 1199102))minus1
]2
(66)
Application of (21) gives the desired inequality (64)
Corollary 7 Assume that the positive numbers 119909 and 119910 arenot equal Then the following inequalities
119875119901119872119901lt 1198602
1199012 (67)
119871119901119879119901lt 1198602
1199012(68)
hold true
Proof For the proof of (67) we let in (64) 1199091= 119866 119909
2= 119860119901
and 1199101= 1199102= 1198601199012
Then the left-hand side of (67) followsfrom (32) and (34) To obtain the right-hand side of theinequality in question let us notice that 119860
119901(119866 119860119901) = 119860
1199012
and obviously 119860119901(1198601199012
1198601199012
) = 1198601199012
Inequality (68) can beestablished in a similar manner We use (64) with 119909
1= 1199092=
1198601199012
1199101= 119866 and 119910
2= 119860119901followed by application of (31)
and (33)
We will close this section proving Wilker-type andHuygens-type inequalities which involve inverse functions ofthe generalized trigonometric and hyperbolic functions Tothis aim we shall employ the following result [23]
Theorem A Let 119906 V 120582 120583 be positive numbers Assume that 119906and V satisfy the separation condition
119906 lt 1 lt V (69)
Then the inequality
1 lt120582
120582 + 120583119906119903+
120583
120582 + 120583V119904 (70)
holds true if either
1 lt 119906120574V120575 119904 gt 0 119903120582 le
119904120583120574
120575(71)
6 International Journal of Mathematics and Mathematical Sciences
or
119906120574V120575 lt 1 119904 lt 0 119903120582 le
119904120583120574
120575 (72)
for some 120574 120575 ge 0with 120574+120575 = 1 If 119906 and V satisfy the separationcondition (69) together with
1 lt 1205741
119906+ 120575
1
V (73)
then inequality (70) is also valid if
119903 le 119904 le minus1 120583120574 le 120582120575 (74)
As in the previous sections the letters 119909 and 119910 will standfor positive and unequal numbers Also 119866119860
1199012and 119860
119901
denote the power means of 119909 and 119910For the sake of notation we define
120591 = 119901119905 where 119905 =1
2log(119909
119910) (75)
We are in a position to prove the following
Theorem 8 Let V119901be the same as in (35) and let 120590
1=
(cosh2(1205912))1119901 Assume that 120582 120583 119904 gt 0 Also let
(119906 V) = (sinminus1119901V119901
tanhminus1119901V119901
1205901
V119901
tanhminus1119901V119901
) (76)
or
(119906 V) = (tanhminus1119901V119901
sinhminus1119901V119901
V119901
sinhminus1119901V119901
) (77)
If
(119901 minus 1) 119903120582 le 119904120583 (78)
then inequality (70) is satisfied if (119906 V) is defined either in (76)or in (77) Inequality (70) is also satisfied if either
(119906 V) = (V119901
sinminus1119901V119901
tanhminus1119901V119901
sinminus1119901V119901
) (79)
or
(119906 V) = (1205902
V119901
tanminus1119901V119901
sinhminus1119901V119901
tanminus1119901V119901
) (80)
provided that
119903120582 le (119901 minus 1) 119904120583 (81)
Here 1205902= 1205901(cosh 120591)1119901
Proof For the proof of validity of (70) with (119906 V) as definedin (76) we let
119906 =119871119901
119875119901
V =119871119901
119866 (82)
It follows from (53) that the separation condition (69) issatisfied To complete the proof of (76) we utilize a well-known fact about bivariatemeans Let119873(119909 119910) equiv 119873 be ameanwhich is homogeneous of degree 1 in its variables Then
This togetherwith (36) and (37) gives the explicit formula (76)for (119906 V) We will show now that the first inequality in (71) issatisfied if 120574 = 1119901 and 120575 = 1 minus 1119901 To this aim we utilize(82) and write inequality (61) as follows
1 lt 119906120572V120573 (85)
where 120572 = 1119901 and 120573 = 1 minus 1119901 This yields 120574 = 120572
and 120575 = 120573 To obtain condition (78) of validity of (70) wesubstitute 120574120575 = 1(119901minus 1) into the last inequality in (71)Thiscompletes the first part of the proof Assume now that (119906 V)is the same as is defined in (79) We will prove that (70) holdstrue provided that condition (81) is satisfied First we define
119906 =119875119901
1198601199012
V =119875119901
119871119901
(86)
Again we appeal to (53) to claim that 119906 and V satisfy theseparation condition (69) Making use of (36) and (37) weobtain an explicit formula (79) for (119906 V)Wewill show that thefirst inequality in (71) is satisfied if 120574 = 1minus1119901 and 120575 = 1119901 Tothis aim we utilize (86) and write inequality (62) as follows
1 lt 119906120573V120572 (87)
where 120572 = 1119901 and 120573 = 1 minus 1119901 To prove that (70) holdstrue if (81) is satisfied we substitute 120574120575 = 119901 minus 1 into the lastinequality in (71) The assertion now follows The remainingtwo cases when (119906 V) is defined in (77) or in (80) can beestablished in the analogous manner In these cases we haveeither
(119906 V) = (119872119901
119879119901
119872119901
1198601199012
) (88)
or
(119906 V) = (119879119901
119860119901
119879119901
119872119901
) (89)
We leave it to the reader to complete the proof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] J M Borwein and P B Borwein Pi and the AGM A Study inAnalytic Number Theory and Computational Complexity JohnWiley amp Sons New York NY USA 1987
[2] B C Carlson ldquoAlgorithms involving arithmetic and geometricmeansrdquoThe American Mathematical Monthly vol 78 pp 496ndash505 1971
[3] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[4] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[5] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[6] B A Bhayo and M Vuorinen ldquoInequalities for eigenfunctionsof the 119901-Laplacianrdquo Issues of Analysis vol 2 no 1 pp 13ndash352013
[7] R Klen M Vuorinen and X Zhang ldquoInequalities for the gen-eralized trigonometric and hyperbolic functionsrdquo Journal ofMathematical Analysis and Applications vol 409 no 1 pp 521ndash529 2014
[8] P Lindqvist ldquoSome remarkable sine and cosine functionsrdquo Ric-erche di Matematica vol 44 no 2 pp 269ndash290 1995
[9] A Baricz B A Bhayo and R Klen ldquoConvexity properties ofgeneralized trigonometric and hyperbolic functionsrdquo Aequa-tiones Mathematicae 2013
[10] B A Bhayo ldquoPowermean inequality of generalized trigonomet-ric functionsrdquo 2012 httparxivorgabs12090873
[11] B A Bhayo and M Vuorinen ldquoOn generalized trigonometricfunctions with two parametersrdquo Journal of ApproximationTheory vol 164 no 10 pp 1415ndash1426 2012
[12] W D Jiang M KWang Y M Chu Y P Jiang and F Qi ldquoCon-vexity of the generalized sine function and the generalized hyp-erbolic sine functionrdquo Journal of ApproximationTheory vol 174pp 1ndash9 2013
[13] E Neuman ldquoInequalities involving inverse circular and inversehyperbolic functionsrdquo Journal of Mathematical Inequalities vol4 no 1 pp 11ndash14 2010
[14] S Takeuchi ldquoGeneralized Jacobian elliptic functions and theirapplication to bifurcation problems associated with 119901-Lapla-cianrdquo Journal of Mathematical Analysis and Applications vol385 no 1 pp 24ndash35 2012
[15] S Takeuchi ldquoThe basis property of generalized Jacobian ellipticfunctionsrdquo 2013 httparxivorgabs13100597
[16] B C Carlson ldquoSome inequalities for hypergeometric func-tionsrdquo Proceedings of the American Mathematical Society vol17 pp 32ndash39 1966
[17] B C Carlson Special Functions of AppliedMathematics Acade-mic Press New York NY USA 1977
[18] B CCarlson ldquoAhypergeometricmean valuerdquoProceedings of theAmerican Mathematical Society vol 16 pp 759ndash766 1965
[19] J L Brenner and B C Carlson ldquoHomogeneous mean valuesweights and asymptoticsrdquo Journal of Mathematical Analysis andApplications vol 123 no 1 pp 265ndash280 1987
[20] F W J Olver D W Lozier R F Boisvert and C W ClarkTheNISTHandbook ofMathematical Functions CambridgeUniver-sity Press New York NY USA 2010
[21] H J Seiffert ldquoProblem 887rdquo Nieuw Archief voor Wiskunde vol11 p 176 1993
[22] H J Seiffert ldquoAufgabe 16rdquoWurzel vol 29 p 87 1995[23] E Neuman ldquoInequalities for weighted sums of powers and their
4 International Journal of Mathematics and Mathematical Sciences
Proof We begin with the proof of (36) Making use of (31)and (22) we obtain
119871119901=
119901radic119860119901
1199012minus 119866119901
coshminus1119901(1198601199012
119866) (40)
Elementary computations yield
119860119901
1199012minus 119866119901= (
1199091199012
+ 1199101199012
2)
2
minus (119909119910)1199012
= (1199091199012
minus 1199101199012
2)
2
(41)
Multiplying and dividing by
(1199091199012
+ 1199101199012
2)
2
(42)
we obtain using (35)
119860119901
1199012minus 119866119901= (1198601199012
V119901)119901
(43)
We shall write the denominator of (40) using (10) and (43) asfollows
coshminus1119901(1198601199012
119866) = tanhminus1
119901
119901radic1 minus (119866
1198601199012
)
119901
= tanhminus1119901
119901radic119860119901
1199012minus 119866119901
119860119901
1199012
= tanhminus1119901V119901
(44)
This in conjunction with (40) and (43) gives the desiredresult (36)
We shall provide now a sketch of the proof of formula(39) It follows from (22) and (34) that
119872119901=
119901radic119860119901
119901 minus 119860119901
1199012
coshminus1119901(1198601199011198601199012
) (45)
Elementary computations yield
119860119901
119901minus 119860119901
1199012= (1198601199012
V119901)119901
(46)
Application of (10) with 119906 = 1198601199011198601199012
givescoshminus1119901(1198601199011198601199012
) = sinhminus1119901V119901 This in conjunction with
(45) and (46) yields the asserted result (39) The remainingtwo formulas for the 119901-versions 119875
119901and 119879
119901of the Seiffert
means can be established in an analogous manner using (32)or (33) (22) and (9) We leave it to the interested reader Theproof is complete
6 Inequalities Involving the SB119901
Means
This section deals with inequalities involving the SB119901means
In particular inequalities for the four means introduced inSection 5 are established Also we shall prove Wilker-typeand Huygens-type inequalities involving inverse functions ofthe generalized trigonometric functions and the generalizedhyperbolic functions
Our first result reads as follows
Theorem 2 Let the positive numbers 119909 and 119910 be such that 119909 gt
119910 Then
119878119861119901(119909 119910) lt 119878119861
119901(119910 119909) (47)
Proof We shall prove the assertion using integral formula(28) and formula (21) Let 119906 gt 1 and let 119905 gt 0 Then 119906
119901gt 1
and
(119905 + 119906119901)1minus1119901
gt (119905 + 1)1minus1119901 (48)
or what is the same that
(119905 + 119906119901)minus1119901
(119905 + 1)minus1
gt (119905 + 119906119901)minus1
(119905 + 1)minus1119901 (49)
because 1 minus 1119901 gt 0 Integration yields
1
119901int
infin
0
(119905 + 119906119901)minus1119901
(119905 + 1)minus1119889119905
gt1
119901int
infin
0
(119905 + 119906119901)minus1
(119905 + 1)minus1119901
119889119905
(50)
or what is the same that the inequality
119877119878(119906119901 1) gt 119877
119878(1 119906119901) (51)
Raising both sides to the power of minus1 and next applyingformula (21) we obtain
SB119901(119906 1) lt SB
119901(1 119906) (52)
Letting 119906 = 119909119910 and next utilizing homogeneity of SB119901 we
obtain the desired result
The four new means defined in Section 5 and the powermeans are comparable We have the following
Proof Thefirst third fourth and the sixth inequalities in (53)follow from their definitions (see (31) (32) (34) and (33))The second and the fifth inequalities can be obtained usingTheorem 2 applied to twopairs of defining equations (31)-(32)and (34)-(33)
Our next result reads as follows
Theorem 4 Let 120572 = 1119901 and 120573 = 1minus1119901 Then the inequality
Applying the permutation symmetry (see (14)) to the R-hypergeometric function on the left-hand side and nextraising both sides of the resulting formula to the power of minus1we obtain
119877minus1119901
(1
119901 1 119910119901 119909119901)
minus1
119910119901minus1
= 119877minus1(1
119901 1 119909119901 119910119901)
minus1
(57)
Since minus1 lt minus1119901 Carlsonrsquos inequality (16) yields
119877minus1(1
119901 1 119909119901 119910119901)
minus1
lt 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus119901
(58)
Combining this with (57) we obtain
119877minus1119901
(1
119901 1 119910119901 119909119901)
minus1
119910119901minus1
lt 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus119901
(59)
Utilizing (20) and (21) we can write the last inequality in theform
SB119901(119910 119909) 119910
119901minus1lt SB119901(119909 119910)
119901
(60)
The desired inequality (54) now follows
Corollary 5 Let the numbers 120572 and 120573 be the same as inTheorem 4 Then the following inequalities
119875120572
119901119866120573lt 119871119901 (61)
119871120572
119901119860120573
1199012lt 119875119901 (62)
119879120572
119901119860120573
1199012lt 119872119901
119872120572
119901119860120573
119901lt 119879119901
(63)
hold true
Proof Inequality (61) can be obtained using Theorem 4 with119909 = 119860
1199012and 119910 = 119866 Next we utilize formulas (31) and (32)
to obtain the desired result In a similar fashion one can proveinequality (62) using Theorem 4 with 119909 = 119866 and 119910 = 119860
1199012
Making use of formulas (31) and (32) yields the assertionTheremaining inequalities (63) can be proven in an analogousmanner We omit further details
Another inequality for the SB119901mean is contained in the
following
Theorem 6 Let 1199091 1199092 1199101 1199102gt 0 Then
next we use (20) and finally we raise both sides of theresulting inequality to the power of minus2 This gives
119877119878(119909119901
1 119910119901
1)minus1
119877119878(119909119901
2 119910119901
2)minus1
le [119877119878(119860119901
119901 (1199091 1199092) 119860119901
119901 (1199101 1199102))minus1
]2
(66)
Application of (21) gives the desired inequality (64)
Corollary 7 Assume that the positive numbers 119909 and 119910 arenot equal Then the following inequalities
119875119901119872119901lt 1198602
1199012 (67)
119871119901119879119901lt 1198602
1199012(68)
hold true
Proof For the proof of (67) we let in (64) 1199091= 119866 119909
2= 119860119901
and 1199101= 1199102= 1198601199012
Then the left-hand side of (67) followsfrom (32) and (34) To obtain the right-hand side of theinequality in question let us notice that 119860
119901(119866 119860119901) = 119860
1199012
and obviously 119860119901(1198601199012
1198601199012
) = 1198601199012
Inequality (68) can beestablished in a similar manner We use (64) with 119909
1= 1199092=
1198601199012
1199101= 119866 and 119910
2= 119860119901followed by application of (31)
and (33)
We will close this section proving Wilker-type andHuygens-type inequalities which involve inverse functions ofthe generalized trigonometric and hyperbolic functions Tothis aim we shall employ the following result [23]
Theorem A Let 119906 V 120582 120583 be positive numbers Assume that 119906and V satisfy the separation condition
119906 lt 1 lt V (69)
Then the inequality
1 lt120582
120582 + 120583119906119903+
120583
120582 + 120583V119904 (70)
holds true if either
1 lt 119906120574V120575 119904 gt 0 119903120582 le
119904120583120574
120575(71)
6 International Journal of Mathematics and Mathematical Sciences
or
119906120574V120575 lt 1 119904 lt 0 119903120582 le
119904120583120574
120575 (72)
for some 120574 120575 ge 0with 120574+120575 = 1 If 119906 and V satisfy the separationcondition (69) together with
1 lt 1205741
119906+ 120575
1
V (73)
then inequality (70) is also valid if
119903 le 119904 le minus1 120583120574 le 120582120575 (74)
As in the previous sections the letters 119909 and 119910 will standfor positive and unequal numbers Also 119866119860
1199012and 119860
119901
denote the power means of 119909 and 119910For the sake of notation we define
120591 = 119901119905 where 119905 =1
2log(119909
119910) (75)
We are in a position to prove the following
Theorem 8 Let V119901be the same as in (35) and let 120590
1=
(cosh2(1205912))1119901 Assume that 120582 120583 119904 gt 0 Also let
(119906 V) = (sinminus1119901V119901
tanhminus1119901V119901
1205901
V119901
tanhminus1119901V119901
) (76)
or
(119906 V) = (tanhminus1119901V119901
sinhminus1119901V119901
V119901
sinhminus1119901V119901
) (77)
If
(119901 minus 1) 119903120582 le 119904120583 (78)
then inequality (70) is satisfied if (119906 V) is defined either in (76)or in (77) Inequality (70) is also satisfied if either
(119906 V) = (V119901
sinminus1119901V119901
tanhminus1119901V119901
sinminus1119901V119901
) (79)
or
(119906 V) = (1205902
V119901
tanminus1119901V119901
sinhminus1119901V119901
tanminus1119901V119901
) (80)
provided that
119903120582 le (119901 minus 1) 119904120583 (81)
Here 1205902= 1205901(cosh 120591)1119901
Proof For the proof of validity of (70) with (119906 V) as definedin (76) we let
119906 =119871119901
119875119901
V =119871119901
119866 (82)
It follows from (53) that the separation condition (69) issatisfied To complete the proof of (76) we utilize a well-known fact about bivariatemeans Let119873(119909 119910) equiv 119873 be ameanwhich is homogeneous of degree 1 in its variables Then
This togetherwith (36) and (37) gives the explicit formula (76)for (119906 V) We will show now that the first inequality in (71) issatisfied if 120574 = 1119901 and 120575 = 1 minus 1119901 To this aim we utilize(82) and write inequality (61) as follows
1 lt 119906120572V120573 (85)
where 120572 = 1119901 and 120573 = 1 minus 1119901 This yields 120574 = 120572
and 120575 = 120573 To obtain condition (78) of validity of (70) wesubstitute 120574120575 = 1(119901minus 1) into the last inequality in (71)Thiscompletes the first part of the proof Assume now that (119906 V)is the same as is defined in (79) We will prove that (70) holdstrue provided that condition (81) is satisfied First we define
119906 =119875119901
1198601199012
V =119875119901
119871119901
(86)
Again we appeal to (53) to claim that 119906 and V satisfy theseparation condition (69) Making use of (36) and (37) weobtain an explicit formula (79) for (119906 V)Wewill show that thefirst inequality in (71) is satisfied if 120574 = 1minus1119901 and 120575 = 1119901 Tothis aim we utilize (86) and write inequality (62) as follows
1 lt 119906120573V120572 (87)
where 120572 = 1119901 and 120573 = 1 minus 1119901 To prove that (70) holdstrue if (81) is satisfied we substitute 120574120575 = 119901 minus 1 into the lastinequality in (71) The assertion now follows The remainingtwo cases when (119906 V) is defined in (77) or in (80) can beestablished in the analogous manner In these cases we haveeither
(119906 V) = (119872119901
119879119901
119872119901
1198601199012
) (88)
or
(119906 V) = (119879119901
119860119901
119879119901
119872119901
) (89)
We leave it to the reader to complete the proof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] J M Borwein and P B Borwein Pi and the AGM A Study inAnalytic Number Theory and Computational Complexity JohnWiley amp Sons New York NY USA 1987
[2] B C Carlson ldquoAlgorithms involving arithmetic and geometricmeansrdquoThe American Mathematical Monthly vol 78 pp 496ndash505 1971
[3] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[4] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[5] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[6] B A Bhayo and M Vuorinen ldquoInequalities for eigenfunctionsof the 119901-Laplacianrdquo Issues of Analysis vol 2 no 1 pp 13ndash352013
[7] R Klen M Vuorinen and X Zhang ldquoInequalities for the gen-eralized trigonometric and hyperbolic functionsrdquo Journal ofMathematical Analysis and Applications vol 409 no 1 pp 521ndash529 2014
[8] P Lindqvist ldquoSome remarkable sine and cosine functionsrdquo Ric-erche di Matematica vol 44 no 2 pp 269ndash290 1995
[9] A Baricz B A Bhayo and R Klen ldquoConvexity properties ofgeneralized trigonometric and hyperbolic functionsrdquo Aequa-tiones Mathematicae 2013
[10] B A Bhayo ldquoPowermean inequality of generalized trigonomet-ric functionsrdquo 2012 httparxivorgabs12090873
[11] B A Bhayo and M Vuorinen ldquoOn generalized trigonometricfunctions with two parametersrdquo Journal of ApproximationTheory vol 164 no 10 pp 1415ndash1426 2012
[12] W D Jiang M KWang Y M Chu Y P Jiang and F Qi ldquoCon-vexity of the generalized sine function and the generalized hyp-erbolic sine functionrdquo Journal of ApproximationTheory vol 174pp 1ndash9 2013
[13] E Neuman ldquoInequalities involving inverse circular and inversehyperbolic functionsrdquo Journal of Mathematical Inequalities vol4 no 1 pp 11ndash14 2010
[14] S Takeuchi ldquoGeneralized Jacobian elliptic functions and theirapplication to bifurcation problems associated with 119901-Lapla-cianrdquo Journal of Mathematical Analysis and Applications vol385 no 1 pp 24ndash35 2012
[15] S Takeuchi ldquoThe basis property of generalized Jacobian ellipticfunctionsrdquo 2013 httparxivorgabs13100597
[16] B C Carlson ldquoSome inequalities for hypergeometric func-tionsrdquo Proceedings of the American Mathematical Society vol17 pp 32ndash39 1966
[17] B C Carlson Special Functions of AppliedMathematics Acade-mic Press New York NY USA 1977
[18] B CCarlson ldquoAhypergeometricmean valuerdquoProceedings of theAmerican Mathematical Society vol 16 pp 759ndash766 1965
[19] J L Brenner and B C Carlson ldquoHomogeneous mean valuesweights and asymptoticsrdquo Journal of Mathematical Analysis andApplications vol 123 no 1 pp 265ndash280 1987
[20] F W J Olver D W Lozier R F Boisvert and C W ClarkTheNISTHandbook ofMathematical Functions CambridgeUniver-sity Press New York NY USA 2010
[21] H J Seiffert ldquoProblem 887rdquo Nieuw Archief voor Wiskunde vol11 p 176 1993
[22] H J Seiffert ldquoAufgabe 16rdquoWurzel vol 29 p 87 1995[23] E Neuman ldquoInequalities for weighted sums of powers and their
Applying the permutation symmetry (see (14)) to the R-hypergeometric function on the left-hand side and nextraising both sides of the resulting formula to the power of minus1we obtain
119877minus1119901
(1
119901 1 119910119901 119909119901)
minus1
119910119901minus1
= 119877minus1(1
119901 1 119909119901 119910119901)
minus1
(57)
Since minus1 lt minus1119901 Carlsonrsquos inequality (16) yields
119877minus1(1
119901 1 119909119901 119910119901)
minus1
lt 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus119901
(58)
Combining this with (57) we obtain
119877minus1119901
(1
119901 1 119910119901 119909119901)
minus1
119910119901minus1
lt 119877minus1119901
(1
119901 1 119909119901 119910119901)
minus119901
(59)
Utilizing (20) and (21) we can write the last inequality in theform
SB119901(119910 119909) 119910
119901minus1lt SB119901(119909 119910)
119901
(60)
The desired inequality (54) now follows
Corollary 5 Let the numbers 120572 and 120573 be the same as inTheorem 4 Then the following inequalities
119875120572
119901119866120573lt 119871119901 (61)
119871120572
119901119860120573
1199012lt 119875119901 (62)
119879120572
119901119860120573
1199012lt 119872119901
119872120572
119901119860120573
119901lt 119879119901
(63)
hold true
Proof Inequality (61) can be obtained using Theorem 4 with119909 = 119860
1199012and 119910 = 119866 Next we utilize formulas (31) and (32)
to obtain the desired result In a similar fashion one can proveinequality (62) using Theorem 4 with 119909 = 119866 and 119910 = 119860
1199012
Making use of formulas (31) and (32) yields the assertionTheremaining inequalities (63) can be proven in an analogousmanner We omit further details
Another inequality for the SB119901mean is contained in the
following
Theorem 6 Let 1199091 1199092 1199101 1199102gt 0 Then
next we use (20) and finally we raise both sides of theresulting inequality to the power of minus2 This gives
119877119878(119909119901
1 119910119901
1)minus1
119877119878(119909119901
2 119910119901
2)minus1
le [119877119878(119860119901
119901 (1199091 1199092) 119860119901
119901 (1199101 1199102))minus1
]2
(66)
Application of (21) gives the desired inequality (64)
Corollary 7 Assume that the positive numbers 119909 and 119910 arenot equal Then the following inequalities
119875119901119872119901lt 1198602
1199012 (67)
119871119901119879119901lt 1198602
1199012(68)
hold true
Proof For the proof of (67) we let in (64) 1199091= 119866 119909
2= 119860119901
and 1199101= 1199102= 1198601199012
Then the left-hand side of (67) followsfrom (32) and (34) To obtain the right-hand side of theinequality in question let us notice that 119860
119901(119866 119860119901) = 119860
1199012
and obviously 119860119901(1198601199012
1198601199012
) = 1198601199012
Inequality (68) can beestablished in a similar manner We use (64) with 119909
1= 1199092=
1198601199012
1199101= 119866 and 119910
2= 119860119901followed by application of (31)
and (33)
We will close this section proving Wilker-type andHuygens-type inequalities which involve inverse functions ofthe generalized trigonometric and hyperbolic functions Tothis aim we shall employ the following result [23]
Theorem A Let 119906 V 120582 120583 be positive numbers Assume that 119906and V satisfy the separation condition
119906 lt 1 lt V (69)
Then the inequality
1 lt120582
120582 + 120583119906119903+
120583
120582 + 120583V119904 (70)
holds true if either
1 lt 119906120574V120575 119904 gt 0 119903120582 le
119904120583120574
120575(71)
6 International Journal of Mathematics and Mathematical Sciences
or
119906120574V120575 lt 1 119904 lt 0 119903120582 le
119904120583120574
120575 (72)
for some 120574 120575 ge 0with 120574+120575 = 1 If 119906 and V satisfy the separationcondition (69) together with
1 lt 1205741
119906+ 120575
1
V (73)
then inequality (70) is also valid if
119903 le 119904 le minus1 120583120574 le 120582120575 (74)
As in the previous sections the letters 119909 and 119910 will standfor positive and unequal numbers Also 119866119860
1199012and 119860
119901
denote the power means of 119909 and 119910For the sake of notation we define
120591 = 119901119905 where 119905 =1
2log(119909
119910) (75)
We are in a position to prove the following
Theorem 8 Let V119901be the same as in (35) and let 120590
1=
(cosh2(1205912))1119901 Assume that 120582 120583 119904 gt 0 Also let
(119906 V) = (sinminus1119901V119901
tanhminus1119901V119901
1205901
V119901
tanhminus1119901V119901
) (76)
or
(119906 V) = (tanhminus1119901V119901
sinhminus1119901V119901
V119901
sinhminus1119901V119901
) (77)
If
(119901 minus 1) 119903120582 le 119904120583 (78)
then inequality (70) is satisfied if (119906 V) is defined either in (76)or in (77) Inequality (70) is also satisfied if either
(119906 V) = (V119901
sinminus1119901V119901
tanhminus1119901V119901
sinminus1119901V119901
) (79)
or
(119906 V) = (1205902
V119901
tanminus1119901V119901
sinhminus1119901V119901
tanminus1119901V119901
) (80)
provided that
119903120582 le (119901 minus 1) 119904120583 (81)
Here 1205902= 1205901(cosh 120591)1119901
Proof For the proof of validity of (70) with (119906 V) as definedin (76) we let
119906 =119871119901
119875119901
V =119871119901
119866 (82)
It follows from (53) that the separation condition (69) issatisfied To complete the proof of (76) we utilize a well-known fact about bivariatemeans Let119873(119909 119910) equiv 119873 be ameanwhich is homogeneous of degree 1 in its variables Then
This togetherwith (36) and (37) gives the explicit formula (76)for (119906 V) We will show now that the first inequality in (71) issatisfied if 120574 = 1119901 and 120575 = 1 minus 1119901 To this aim we utilize(82) and write inequality (61) as follows
1 lt 119906120572V120573 (85)
where 120572 = 1119901 and 120573 = 1 minus 1119901 This yields 120574 = 120572
and 120575 = 120573 To obtain condition (78) of validity of (70) wesubstitute 120574120575 = 1(119901minus 1) into the last inequality in (71)Thiscompletes the first part of the proof Assume now that (119906 V)is the same as is defined in (79) We will prove that (70) holdstrue provided that condition (81) is satisfied First we define
119906 =119875119901
1198601199012
V =119875119901
119871119901
(86)
Again we appeal to (53) to claim that 119906 and V satisfy theseparation condition (69) Making use of (36) and (37) weobtain an explicit formula (79) for (119906 V)Wewill show that thefirst inequality in (71) is satisfied if 120574 = 1minus1119901 and 120575 = 1119901 Tothis aim we utilize (86) and write inequality (62) as follows
1 lt 119906120573V120572 (87)
where 120572 = 1119901 and 120573 = 1 minus 1119901 To prove that (70) holdstrue if (81) is satisfied we substitute 120574120575 = 119901 minus 1 into the lastinequality in (71) The assertion now follows The remainingtwo cases when (119906 V) is defined in (77) or in (80) can beestablished in the analogous manner In these cases we haveeither
(119906 V) = (119872119901
119879119901
119872119901
1198601199012
) (88)
or
(119906 V) = (119879119901
119860119901
119879119901
119872119901
) (89)
We leave it to the reader to complete the proof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] J M Borwein and P B Borwein Pi and the AGM A Study inAnalytic Number Theory and Computational Complexity JohnWiley amp Sons New York NY USA 1987
[2] B C Carlson ldquoAlgorithms involving arithmetic and geometricmeansrdquoThe American Mathematical Monthly vol 78 pp 496ndash505 1971
[3] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[4] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[5] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[6] B A Bhayo and M Vuorinen ldquoInequalities for eigenfunctionsof the 119901-Laplacianrdquo Issues of Analysis vol 2 no 1 pp 13ndash352013
[7] R Klen M Vuorinen and X Zhang ldquoInequalities for the gen-eralized trigonometric and hyperbolic functionsrdquo Journal ofMathematical Analysis and Applications vol 409 no 1 pp 521ndash529 2014
[8] P Lindqvist ldquoSome remarkable sine and cosine functionsrdquo Ric-erche di Matematica vol 44 no 2 pp 269ndash290 1995
[9] A Baricz B A Bhayo and R Klen ldquoConvexity properties ofgeneralized trigonometric and hyperbolic functionsrdquo Aequa-tiones Mathematicae 2013
[10] B A Bhayo ldquoPowermean inequality of generalized trigonomet-ric functionsrdquo 2012 httparxivorgabs12090873
[11] B A Bhayo and M Vuorinen ldquoOn generalized trigonometricfunctions with two parametersrdquo Journal of ApproximationTheory vol 164 no 10 pp 1415ndash1426 2012
[12] W D Jiang M KWang Y M Chu Y P Jiang and F Qi ldquoCon-vexity of the generalized sine function and the generalized hyp-erbolic sine functionrdquo Journal of ApproximationTheory vol 174pp 1ndash9 2013
[13] E Neuman ldquoInequalities involving inverse circular and inversehyperbolic functionsrdquo Journal of Mathematical Inequalities vol4 no 1 pp 11ndash14 2010
[14] S Takeuchi ldquoGeneralized Jacobian elliptic functions and theirapplication to bifurcation problems associated with 119901-Lapla-cianrdquo Journal of Mathematical Analysis and Applications vol385 no 1 pp 24ndash35 2012
[15] S Takeuchi ldquoThe basis property of generalized Jacobian ellipticfunctionsrdquo 2013 httparxivorgabs13100597
[16] B C Carlson ldquoSome inequalities for hypergeometric func-tionsrdquo Proceedings of the American Mathematical Society vol17 pp 32ndash39 1966
[17] B C Carlson Special Functions of AppliedMathematics Acade-mic Press New York NY USA 1977
[18] B CCarlson ldquoAhypergeometricmean valuerdquoProceedings of theAmerican Mathematical Society vol 16 pp 759ndash766 1965
[19] J L Brenner and B C Carlson ldquoHomogeneous mean valuesweights and asymptoticsrdquo Journal of Mathematical Analysis andApplications vol 123 no 1 pp 265ndash280 1987
[20] F W J Olver D W Lozier R F Boisvert and C W ClarkTheNISTHandbook ofMathematical Functions CambridgeUniver-sity Press New York NY USA 2010
[21] H J Seiffert ldquoProblem 887rdquo Nieuw Archief voor Wiskunde vol11 p 176 1993
[22] H J Seiffert ldquoAufgabe 16rdquoWurzel vol 29 p 87 1995[23] E Neuman ldquoInequalities for weighted sums of powers and their
6 International Journal of Mathematics and Mathematical Sciences
or
119906120574V120575 lt 1 119904 lt 0 119903120582 le
119904120583120574
120575 (72)
for some 120574 120575 ge 0with 120574+120575 = 1 If 119906 and V satisfy the separationcondition (69) together with
1 lt 1205741
119906+ 120575
1
V (73)
then inequality (70) is also valid if
119903 le 119904 le minus1 120583120574 le 120582120575 (74)
As in the previous sections the letters 119909 and 119910 will standfor positive and unequal numbers Also 119866119860
1199012and 119860
119901
denote the power means of 119909 and 119910For the sake of notation we define
120591 = 119901119905 where 119905 =1
2log(119909
119910) (75)
We are in a position to prove the following
Theorem 8 Let V119901be the same as in (35) and let 120590
1=
(cosh2(1205912))1119901 Assume that 120582 120583 119904 gt 0 Also let
(119906 V) = (sinminus1119901V119901
tanhminus1119901V119901
1205901
V119901
tanhminus1119901V119901
) (76)
or
(119906 V) = (tanhminus1119901V119901
sinhminus1119901V119901
V119901
sinhminus1119901V119901
) (77)
If
(119901 minus 1) 119903120582 le 119904120583 (78)
then inequality (70) is satisfied if (119906 V) is defined either in (76)or in (77) Inequality (70) is also satisfied if either
(119906 V) = (V119901
sinminus1119901V119901
tanhminus1119901V119901
sinminus1119901V119901
) (79)
or
(119906 V) = (1205902
V119901
tanminus1119901V119901
sinhminus1119901V119901
tanminus1119901V119901
) (80)
provided that
119903120582 le (119901 minus 1) 119904120583 (81)
Here 1205902= 1205901(cosh 120591)1119901
Proof For the proof of validity of (70) with (119906 V) as definedin (76) we let
119906 =119871119901
119875119901
V =119871119901
119866 (82)
It follows from (53) that the separation condition (69) issatisfied To complete the proof of (76) we utilize a well-known fact about bivariatemeans Let119873(119909 119910) equiv 119873 be ameanwhich is homogeneous of degree 1 in its variables Then
This togetherwith (36) and (37) gives the explicit formula (76)for (119906 V) We will show now that the first inequality in (71) issatisfied if 120574 = 1119901 and 120575 = 1 minus 1119901 To this aim we utilize(82) and write inequality (61) as follows
1 lt 119906120572V120573 (85)
where 120572 = 1119901 and 120573 = 1 minus 1119901 This yields 120574 = 120572
and 120575 = 120573 To obtain condition (78) of validity of (70) wesubstitute 120574120575 = 1(119901minus 1) into the last inequality in (71)Thiscompletes the first part of the proof Assume now that (119906 V)is the same as is defined in (79) We will prove that (70) holdstrue provided that condition (81) is satisfied First we define
119906 =119875119901
1198601199012
V =119875119901
119871119901
(86)
Again we appeal to (53) to claim that 119906 and V satisfy theseparation condition (69) Making use of (36) and (37) weobtain an explicit formula (79) for (119906 V)Wewill show that thefirst inequality in (71) is satisfied if 120574 = 1minus1119901 and 120575 = 1119901 Tothis aim we utilize (86) and write inequality (62) as follows
1 lt 119906120573V120572 (87)
where 120572 = 1119901 and 120573 = 1 minus 1119901 To prove that (70) holdstrue if (81) is satisfied we substitute 120574120575 = 119901 minus 1 into the lastinequality in (71) The assertion now follows The remainingtwo cases when (119906 V) is defined in (77) or in (80) can beestablished in the analogous manner In these cases we haveeither
(119906 V) = (119872119901
119879119901
119872119901
1198601199012
) (88)
or
(119906 V) = (119879119901
119860119901
119879119901
119872119901
) (89)
We leave it to the reader to complete the proof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
International Journal of Mathematics and Mathematical Sciences 7
References
[1] J M Borwein and P B Borwein Pi and the AGM A Study inAnalytic Number Theory and Computational Complexity JohnWiley amp Sons New York NY USA 1987
[2] B C Carlson ldquoAlgorithms involving arithmetic and geometricmeansrdquoThe American Mathematical Monthly vol 78 pp 496ndash505 1971
[3] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[4] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[5] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[6] B A Bhayo and M Vuorinen ldquoInequalities for eigenfunctionsof the 119901-Laplacianrdquo Issues of Analysis vol 2 no 1 pp 13ndash352013
[7] R Klen M Vuorinen and X Zhang ldquoInequalities for the gen-eralized trigonometric and hyperbolic functionsrdquo Journal ofMathematical Analysis and Applications vol 409 no 1 pp 521ndash529 2014
[8] P Lindqvist ldquoSome remarkable sine and cosine functionsrdquo Ric-erche di Matematica vol 44 no 2 pp 269ndash290 1995
[9] A Baricz B A Bhayo and R Klen ldquoConvexity properties ofgeneralized trigonometric and hyperbolic functionsrdquo Aequa-tiones Mathematicae 2013
[10] B A Bhayo ldquoPowermean inequality of generalized trigonomet-ric functionsrdquo 2012 httparxivorgabs12090873
[11] B A Bhayo and M Vuorinen ldquoOn generalized trigonometricfunctions with two parametersrdquo Journal of ApproximationTheory vol 164 no 10 pp 1415ndash1426 2012
[12] W D Jiang M KWang Y M Chu Y P Jiang and F Qi ldquoCon-vexity of the generalized sine function and the generalized hyp-erbolic sine functionrdquo Journal of ApproximationTheory vol 174pp 1ndash9 2013
[13] E Neuman ldquoInequalities involving inverse circular and inversehyperbolic functionsrdquo Journal of Mathematical Inequalities vol4 no 1 pp 11ndash14 2010
[14] S Takeuchi ldquoGeneralized Jacobian elliptic functions and theirapplication to bifurcation problems associated with 119901-Lapla-cianrdquo Journal of Mathematical Analysis and Applications vol385 no 1 pp 24ndash35 2012
[15] S Takeuchi ldquoThe basis property of generalized Jacobian ellipticfunctionsrdquo 2013 httparxivorgabs13100597
[16] B C Carlson ldquoSome inequalities for hypergeometric func-tionsrdquo Proceedings of the American Mathematical Society vol17 pp 32ndash39 1966
[17] B C Carlson Special Functions of AppliedMathematics Acade-mic Press New York NY USA 1977
[18] B CCarlson ldquoAhypergeometricmean valuerdquoProceedings of theAmerican Mathematical Society vol 16 pp 759ndash766 1965
[19] J L Brenner and B C Carlson ldquoHomogeneous mean valuesweights and asymptoticsrdquo Journal of Mathematical Analysis andApplications vol 123 no 1 pp 265ndash280 1987
[20] F W J Olver D W Lozier R F Boisvert and C W ClarkTheNISTHandbook ofMathematical Functions CambridgeUniver-sity Press New York NY USA 2010
[21] H J Seiffert ldquoProblem 887rdquo Nieuw Archief voor Wiskunde vol11 p 176 1993
[22] H J Seiffert ldquoAufgabe 16rdquoWurzel vol 29 p 87 1995[23] E Neuman ldquoInequalities for weighted sums of powers and their
International Journal of Mathematics and Mathematical Sciences 7
References
[1] J M Borwein and P B Borwein Pi and the AGM A Study inAnalytic Number Theory and Computational Complexity JohnWiley amp Sons New York NY USA 1987
[2] B C Carlson ldquoAlgorithms involving arithmetic and geometricmeansrdquoThe American Mathematical Monthly vol 78 pp 496ndash505 1971
[3] E Neuman ldquoInequalities for the Schwab-Borchardt mean andtheir applicationsrdquo Journal of Mathematical Inequalities vol 5no 4 pp 601ndash609 2011
[4] E Neuman and J Sandor ldquoOn the Schwab-Borchardt meanrdquoMathematica Pannonica vol 14 no 2 pp 253ndash266 2003
[5] E Neuman and J Sandor ldquoOn the Schwab-Borchardt mean IIrdquoMathematica Pannonica vol 17 no 1 pp 49ndash59 2006
[6] B A Bhayo and M Vuorinen ldquoInequalities for eigenfunctionsof the 119901-Laplacianrdquo Issues of Analysis vol 2 no 1 pp 13ndash352013
[7] R Klen M Vuorinen and X Zhang ldquoInequalities for the gen-eralized trigonometric and hyperbolic functionsrdquo Journal ofMathematical Analysis and Applications vol 409 no 1 pp 521ndash529 2014
[8] P Lindqvist ldquoSome remarkable sine and cosine functionsrdquo Ric-erche di Matematica vol 44 no 2 pp 269ndash290 1995
[9] A Baricz B A Bhayo and R Klen ldquoConvexity properties ofgeneralized trigonometric and hyperbolic functionsrdquo Aequa-tiones Mathematicae 2013
[10] B A Bhayo ldquoPowermean inequality of generalized trigonomet-ric functionsrdquo 2012 httparxivorgabs12090873
[11] B A Bhayo and M Vuorinen ldquoOn generalized trigonometricfunctions with two parametersrdquo Journal of ApproximationTheory vol 164 no 10 pp 1415ndash1426 2012
[12] W D Jiang M KWang Y M Chu Y P Jiang and F Qi ldquoCon-vexity of the generalized sine function and the generalized hyp-erbolic sine functionrdquo Journal of ApproximationTheory vol 174pp 1ndash9 2013
[13] E Neuman ldquoInequalities involving inverse circular and inversehyperbolic functionsrdquo Journal of Mathematical Inequalities vol4 no 1 pp 11ndash14 2010
[14] S Takeuchi ldquoGeneralized Jacobian elliptic functions and theirapplication to bifurcation problems associated with 119901-Lapla-cianrdquo Journal of Mathematical Analysis and Applications vol385 no 1 pp 24ndash35 2012
[15] S Takeuchi ldquoThe basis property of generalized Jacobian ellipticfunctionsrdquo 2013 httparxivorgabs13100597
[16] B C Carlson ldquoSome inequalities for hypergeometric func-tionsrdquo Proceedings of the American Mathematical Society vol17 pp 32ndash39 1966
[17] B C Carlson Special Functions of AppliedMathematics Acade-mic Press New York NY USA 1977
[18] B CCarlson ldquoAhypergeometricmean valuerdquoProceedings of theAmerican Mathematical Society vol 16 pp 759ndash766 1965
[19] J L Brenner and B C Carlson ldquoHomogeneous mean valuesweights and asymptoticsrdquo Journal of Mathematical Analysis andApplications vol 123 no 1 pp 265ndash280 1987
[20] F W J Olver D W Lozier R F Boisvert and C W ClarkTheNISTHandbook ofMathematical Functions CambridgeUniver-sity Press New York NY USA 2010
[21] H J Seiffert ldquoProblem 887rdquo Nieuw Archief voor Wiskunde vol11 p 176 1993
[22] H J Seiffert ldquoAufgabe 16rdquoWurzel vol 29 p 87 1995[23] E Neuman ldquoInequalities for weighted sums of powers and their
