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On nk

N,q ; nSummability Factors of Infinite

Series (III) Aradhana Dutt Jauhari

ABSTRACT: A theorem concerning some new absolute summability method is proved. Many other results some known and unknown are derived.

KEY WORDS AND PHRASES: Absolute Summability, Summability.

—————————— ——————————

1- INTRODUCTION :

Let na be a given infinite series with partial sumsns and n nu na . Let

n and

nt be the thn Cesàro mean of

order ( >-1)of the sequencens and

nu respectevily.The series na is said to be summable k

C, ,k 1 ,if

kk 1

n n 1

n 1

n (1.1)

Or equivalently,

k1

n

n 1

n t

where,

n n n 1t n

A series na is summable k

C, ; n , if the series

kk k 1

n n 1

n 1

n n ; (1.2)

(n) , a positive non-decreasing sequence such that nm n m .

where n

is the thn Cesàro mean of order of na ,It follows that above definition reduces to that of FLETT [4].

Let np be a sequence of positive numbers such that,

n

n v

v 0

P p

as n

.

The sequence -to -sequence transformation

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n

n v v

v 0n

1T p s

P

nP 0(n 0) (1.3) defines the sequence of weighted means of the

sequencens generated by the sequence of coefficients

np . The series na is said to be summable nk

N,p ,k 1 if

Email id- aditya_jauhari@ rediffmail.com, Deptt. Of Applied Sciences (Mathematics), N.I.E.T., Greater Noida ,U.P. (INDIA)

kk 1

nn n 1

n 1 n

Pu u

p (1.4)

and it is said to be summable nk

N,p ; n ,k 1 ,if

kk k 1

n nn n 1

n 1 n n

P PT T

p p ; (1.5)

In 1995, SULAIMAN, W.T. [8] proved the theorem.The main objective of this paper is to generalize the theorem of SULAIMAN[8]. However our theorem

is as follows. 2-THEOREM:

Let { np } and { nq } be two sequences of positive real numbers such that-

n 1 np (p ) (2.1)

n n 1q (q ) (2.2)

n n n np Q (P q ) (2.3)

The series na is summable n

kN,p , n then n na is summable n

kN,q , n if –

1/k

n n n n nn

n n n n n

Q p Q P q

q P q p Q (2.4)

1/k

n n n nn

n n n n

p Q P q

P q p Q (2.5)

3-PROOF OF THE THEOREM:

In order to prove the theorem it is sufficient to prove that

by LEINDLER [5]

kk k 1

n nn n 1

n 1 n n

P PT T

p p

where ,

n

n v v

v 0n

1T p s

P

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Let n n n 1x T T , n 1 0 0,s a

nn

n v 1 v

v 1n n 1

px P a

P P

nn n 1

n v 1 v

v 1n

P Px P a

p

n 1 n 2

n n 1

n 1 n 1

1 P Pa x

P p

Considering-

n v

n v

v 0 0n

1t q a

Q

n 1 n

n v v 1 v 1 n v v

v 0 v 0n

1t Q ( a ) Q a

Q

n 1 n

v v 1 v 1 v v

v 0 v 0n

1Q (a ) a

Q (3.1)

so,

n 1

n n 1 v v 1 v 1

v 0n

n 2 n 1 n

v v 1 v 1 v v v v

v 0 v 0 v 0n 1

1t t Q (a )

Q

1 Q (a ) a a

Q

n 1 n 2

v v 1 v 1 v v 1 v 1

v 0 v 0n n 1

1 1Q (a ) Q (a )

Q Q

n 1v

v v 1 v 1

v 0n n 1

qQ (a )

Q Q

n 1v

v 1 v v

v 0n n 1

qQ a

Q Q

nv n 1 n 2

v 1 v n 1

v 0n n 1 n 1 n 1

q 1 P PQ x

Q Q P p

(3.2)

Using Abel’s transformation, we have-

n 1 v1 2v 1 vn

n n 1 1

v 0 0n n 1 v 1 1

P PQqt t x

Q Q P p+

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nn 1 n v 1 v 2

v 1

v 0n 1 v 1

Q P Px

P p

n 1n v 1 v v v 1

v

v 0n n 1 v 1 v

n 1 n n n 1n

n 1 n

q Q P Px

Q Q P p

Q P P x

P p

Clearly,

v 1 v v v 1 v v 1

v 1 v v

v 1 v v 1 v v

Q p Q qQ

P P P P P

so,

n 1 n 1n n v 1

n n 1 v 1 v v v v 1 v

v 0 v 0n n 1 n n 1 v

q q Pt t Q x x Q

Q Q Q Q p

n 1n v 1 n n

v v v 1 n n

v 0n n 1 v n n

q P q Px q x

Q Q p Q p

=1 2 3 4

(say)

Now for n na to be summable nk

N,q , n if-

kk k 1

n nn n 1

n 1 n n

Q QT T

q q (3.3)

Now by Minköwaski’s inequality, we have-

kk k 1

n n

1 2 3 4n 1 n n

Q Q

q q

k k 1k

n n

1n 1 n n

k k 1 k k 1kk

n n n n

2 3n 1 n 1n n n n

Q QM

q q

Q Q Q Q +

q q q q

k k 1k

n n

4n 1 n n

Q Q

q q

where M is some positive constant.

=11 12 13 14

(say)

Now,

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k k 1k

v v

11 1v 1 v v

Q QM

q q

k k 1 kn 1

v v nv 1 v v

v 1 v 0v v n n 1

Q Q qM Q x

q q Q Q

kk k 1k n 1

v 1 vn n n nv vk 1 k k

n 1 v 0n n n n n 1 v

Q qQ Q q 1 qx

q q q Q Q q

kk n 1k 1 k n 1 n 1k kv 1n n n

v v v vk 1 k kn 1 v 0 v 0n n n n 1 v

QQ Q qx q q

q q Q Q q

Using Hölder’s inequality-

k k 1 k 1kn 1

k kv v vn nv v v

n 1 v 0n n n 1 v v v

Q P pQ qx q

q Q Q q p P

k k 1 kn 1

k kn v v v v vv v v

n 1 v 0n n 1 v v v v v

q Q P Q p Px q

Q Q q p q P p

k k 1n 1

k kv v v nv v v

v 1 n v 1v v v n n 1

Q P P qx q

q p p Q Q v v v vp Q P q

k k 1n 1

k kv v v vv v

v 0 v v v v

Q P P qx

q p Q p

k k 1 kn 1

kv v v v v v v vv

v 0 v v v v v v v v

Q P P q Q p P qx

q p Q p P q Q p

k k 1n 1

kv vv

v 0 v v

P Px

p p

1

Again

k k 1k

v v

12 2v 1 v v

Q QM

q q

kk k 1n 1

n n n v 1v 1 v v

v 1 v 0n n n n 1 v

Q Q q PM Q x

q q Q Q p

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kk k 1k n 1

n n n v 1v 1 v vk k

n 1 v 0n n n n 1 v

Q Q q PQ x

q q Q Q p

kk

n 1n n v 1 v 1 v

v vkn 1 v 0n n n 1 v v

Q q P Q qx

q Q Q p q

k k kn 1

k kv v 1 v 1nv v vk

n 1 v 0n n 1 v v v

k 1n 1

v

v 0

Q P Qqq x *

Q Q q p q

* q

Using Hölder’s inequality-

k k kk 1 n 1

k kv v vn n 1v v vk

n 1 v 0n n 1 v v v

Q Q Pq Qq x

Q Q q q p

k 1 k kn 1

k kv v v vnv v v

n 1 v 0n n 1 v v v v

P Q Q Pqx q

Q Q p q q p

k 1 k k

k kv v v v nv v v

v 1 n v 1v v v v n n 1

P P Q Q qx q

p p q q Q Q

k 1 k k

kv v v v v v v vv

v 1 v v v v v v v v

P P q Q p Q P qx

p p Q q q P p Q

{using condition (2.4) of theorem }

k k 1

kv vv

v 1 v v

P Px

p p

(1)

Further,

k k 1k

v v

13 3v 1 v v

Q QM

q q

kk k 1n 1

v v v 1nv v v 1

v 1 v 0v v n n 1 v

Q Q PqM x q

q q Q Q p

k k 1 kk n 1

v v v 1nv v v 1k k

v 1 v 0v v n n 1 v

Q Q PqM x q

q q Q Q p

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kk k 1n 1 n 1

k kv 1n nv v v 1 vk

n 1 v 0 v 0n n n 1 v

PQ qx q q

q Q Q p

kkk 1 n 1

k kv 1n n n 1v v v 1k

n 1 v 0n n n 1 v

PQ q Qx q

q Q Q p

kkn 1

k kv 1n nv v v 1

n 1 v 0n n n 1 v

PQ qx q

q Q Q p

k 1k

k kv vn nv v v 1

v 1 n v 1n v v n n 1

P PQ qx q

q p p Q Q

k k 1

k kv v v vv v 1

v 1 v v v v

Q P P qx

q p Q p

k k 1 k

kv v v v v 1 v 1 v 1 v 1v

v 1 v v v v v 1 v 1 v 1 v 1

Q P P q p Q p Qx

q p p Q P q P q

{using condition (2.5) of theorem}

k k 1 k k

kv v v 1 v 1v

v 1 v v v 1 v 1

Q P P qx

q p p Q

k k 1

kv vv

v 1 v v

P Px

p p

1

Lastly,

k k 1k

v v

14 4v 1 v v

Q QM

q q

k k 1 k

v v v vv v

v 1 v v v v

Q Q q PM x

q q Q p

k k 1k k

k kn n n nn nk k

n 1 n n n n

Q Q q Px

q q Q p

k kk

kn n n n n n nnk

n 1 n n n n n n n

Q q P p Q P qx

q Q p q P Q p

k k 1

kn nn

n 1 n n

P Px

p p

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1

This completes the proof of theorem.

4 -COROLLARIES :

The following corollaries can be derived from the theorem- Cor.1:

In the special case when n n our theorem reduces to the theorem of SINHA and KUMAR [10].

Cor.2:

If n 0 for every n , then our theorem reduces to theorem of SHARMA[8].

Cor.3:

For n 0 and k=1 our theorem reduces to the following theorem –

Let { np } and { nq } be two sequences of positive real numbers such that-

n 1 np (p ) (4.1)

n n 1q (q ) (4.2)

n n n np Q (P q ) (4.3)

If na is summable n

kN,p then n na is summable n

kN,q if –

n n n

n

n n n

Q p Q

q P q (4.4)

n n

n

n n

p Q

P q (4.5)

Acknowledgement: I am very thankful to Dr. Rajiv Sinha (Associate Professor, S.M.P.G. College ,Chandausi ,U.P., India ) ,whose great inspiration lead me to complete this paper.

REFERENCES

[1] BOR,H.- On two summability methods ;Math. Proc.Cambridge Phil.Soc.97,(1985),147-149.

[2] BOR,H. and THORPE,B.-On some absolute summability methods;Analysis 7,(1987),145-152.

[3] BORWEIN D. and CASS,F.P.-Strong Nölund summability Math Zeith,103,(1968) 91-111.

[4] FLETT,T.M.-Proc.London Math. Soc.7,(1957),113-141.

[5] LEINDLER,L.-Acta math. Hungar 64 (1994) 269-281.

[6] MAZHAR S.M. – On k

C, summability factors of infinite series , Extract du Bulletin de 1`Academic royale de Belgeque 9classe des sciences

)senance du Samedi 6 mars (1971).

[7] MOHAPTRA, R.N.-A note on summability factors, J. Indian Math. Soc. (1967), 213-224 .

[8] SHARMA,N. –Some aspect of weighted mean matrices,Ph.D. Theses,Kurushetra University (2000).

[9] SINHA,P. and KUMAR,H.– A note on nk

N,p ; summability factors of infinite series factors of infinite series , Tamkang J.of Mathematics ,

vol.39, Number 3,193-198 Autumn 2008.

[10]__________-International Journal of Math., (Communicated).

[11] SULAIMAN, W.T.-Inclusion theorem for absolute matrix summability method of infinite series ; Int. Math. Forum,4,no.24 (2009),1181-1189.

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