ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2013, Vol. 34, No. 3, pp. 272–281. c© Pleiades Publishing, Ltd., 2013.

Approximation by the Durrmeyer-Baskakov-Stancu Operators

Vishnu Narayan Mishra* and Prashantkumar Patel**

(Submitted by F. G. Avkhadiev)

Department of Applied Mathematics and Humanities,Sardar Vallabhbhai National Institute of Technology,

Ichchhanath Mahadev Road, Surat, Surat-395 007 (Gujarat), IndiaReceived November 17, 2012; in final form, March 22, 2013

Abstract—In the present paper we propose the Durrmeyer-Baskakov-Stancu operators. We es-tablish some direct results in the polynomial weighted space of continuous functions defined on theinterval [0,∞). Also, Voronovskaja type theorem is studied.

DOI: 10.1134/S1995080213030074

Keywords and phrases: Durrmeyer type operators, Weighted approximation, Rate of conver-gence, Stancu operators, Modulus of continuities.

1. INTRODUCTION

For f ∈ C[0,∞), the Durrmeyer-Baskakov operators were study in [1] is define as

Dn(f, x) = (n − 1)∞∑

k=0

pn,k(x)

∞∫

0

pn,k(t)f(t)dt, (1.1)

where pn,k(x) =(

n + k − 1k

)xk

(1 + x)n+k.

In [2] Stancu introduce the following generalization of Bernstein polynomials

Sαn (f, x) =

n∑

k=0

f

(k

n

)P k

n,α(x), 0 ≤ x ≤ 1, (1.2)

where Pnn,α(x) =

(n

k

)∏k−1s=0(x + αs)

∏n−k−1s=0 (1 − x + αs)

∏n−1s=0 (1 + αs)

. We get the classical Bernstein polynomials

by putting α = 0. Starting with two parameter α, β satisfying the condition 0 ≤ α ≤ β. In 1983, theother generalization of Stancu opreators was given in[3] and studied the linear positive operatorsSα,β

n : C[0, 1] → C[0, 1] defined for any f ∈ C[0, 1] as follows:

Sα,βn (f, x) =

n∑

k=0

p′n,k(x)f(

k + α

n + β

), 0 ≤ x ≤ 1, (1.3)

where p′n,k(x) =(

n

k

)xk(1 − x)n−k is the Bernstein basis function(cf. [4]).

*E-mail: vishnunarayanmishra@gmail.com**E-mail: prashant225@gmail.com

272

APPROXIMATION BY THE DURRMEYER-BASKAKOV-STANCU OPERATORS 273

Recently, Ibrahim [5] introduced Stancu-Chlodowsky polynomial and investigated convergence andapproximation properties of these operators. Motivated by such type operators we generalize Stancutype generalization of the Durrmeyer type modification of the Baskakov type operators as follows:

D(α,β)n (f, x) = (n − 1)

∞∑

k=0

pn,k(x)

∞∫

0

pn,k(t)f(

nt + α

n + β

)dt, (1.4)

where pn,k(x) define as same in (1.1). The operators D(α,β)n (f, x) in (1.4) are called Durrmeyer-

Baskakov-Stancu operators. For α = 0, β = 0 the operators (1.4) reduce to the operators (1.1).We know that

∞∑

k=0

pn,k(x) = 1,

∞∫

0

pn,k(x) =1

n − 1.

The aim of this paper is to study the approximation properties of the Durrmeyer type modification ofthe Baskakov-Stancu type operators. We estimate moments for these operators. Also, we study directtheorem, rate of approximation, Voronovskaja type asymptotic formula for these operators and weightedapproximation properties for these operators. Finally, we give better error estimations for the operators

D(α,β)n .

2. MOMENT ESTIMATE

Lemma 1. [1] The following equality holds:

Dn(1, x) = 1, Dn(t, x) =1 + nx

n − 2, for n > 2,

Dn(t2, x) =(n2 + n)x2 + 4nx + 2

(n − 3)(n − 2), for n > 3.

Lemma 2. The following equality holds:

D(α,β)n (1, x) = 1, D(α,β)

n (t, x) =n2x + n + α(n − 2)

(n − 2)(n + β), for n > 2,

D(α,β)n (t2, x) =

(n + 1)n3x2 + (4n + 2α(n − 3))n2x + (2 + α2 + 2α)n2 − (6α + 5α2)n + 6α2

(n − 2)(n − 3)(n + β)2,

for n > 3.

Proof. Observer that, D(α,β)n (1, x) = Dn(1, x) = 1,

D(α,β)n (t, x) =

n

n + βDn(t, x) +

α

n + βDn(1, x)

=n

n + β

[1 + nx

n − 2

]+

α

n + β=

n2x + n + α(n − 2)(n − 2)(n + β)

.

Finally for n > 3,

D(α,β)n (t2, x) =

n2

(n + β)2Dn(t2, x) +

2nα

(n + β)2Dn(t, x) +

α2

(n + β)2Dn(1, x)

=(n + 1)n3x2

(n − 2)(n − 3)(n + β)2+

(4n3

(n − 2)(n − 3)(n + β)2+

2n2α

(n − 2)(n + β)2

)x

+2n2

(n − 2)(n − 3)(n + β)2+

2nα

(n − 2)(n + β)2+

α2

(n + β)2

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274 MISHRA, PATEL

=(n + 1)n3x2 + (4n + 2α(n − 3))n2x + (2 + α2 + 2α)n2 − (6α + 5α2)n + 6α2

(n − 2)(n − 3)(n + β)2.

�

Remark 1. For all m ∈ {0, 1, 2, ...}, 0 ≤ α ≤ β; we have the following recursive relation for the

images of the monomials tm under D(α,β)n (tm, x) in terms of Dn(tj , x), j = 0, 1, 2, ...,m, as

D(α,β)n (tm, x) =

m∑

j=0

(m

j

)njαm−j

(n + β)mDn(tj , x).

Remark 2. By simple computation, we have

D(α,β)n (t − x, x) =

(2n + 2β − nβ)x + (1 + α)n − 2α(n − 2)(n + β)

, n > 2,

D(α,β)n ((t − x)2, x) = D(α,β)

n (t2, x) − 2xD(α,β)n (t, x) + x2D(α,β)

n (1, x)

=(

2n3 + (β2 − 4β + 6)n2 + (12β − 5β2)n + 6β2

(n − 2)(n − 3)(n + β)2

)x2

+(

2n3 + (6 + 4α − 2β − 2αβ)n2 + (6β + 10αβ − 12α)n − 12αβ

(n − 2)(n − 3)(n + β)2

)x

+(2 + α2 + 2α)n2 − (6α + 5α2)n + 6α2

(n − 2)(n − 3)(n + β)2.

Lemma 3. For n > 3 be given number, we have

D(α,β)n ((t − x)2, x) ≤ 7β2 + 12β + 8

n − 2

(φ2(x) +

1n − 3

),

where φ2(x) = x(1 + x), x ∈ [0,∞).

Proof. Proceeding along the line of the estimate for D(α,β)n ((t − x)2, x), proof of the lemma follows.

D(α,β)n ((t − x)2, x) =

(2n3 + (β2 − 4β + 6)n2 + (12β − 5β2)n + 6β2

(n − 2)(n − 3)(n + β)2

)x2

+(

2n3 + (6 + 4α − 2β − 2αβ)n2 + (6β + 10αβ − 12α)n − 12αβ

(n − 2)(n − 3)(n + β)2

)x

+(2 + α2 + 2α)n2 − (6α + 5α2)n + 6α2

(n − 2)(n − 3)(n + β)2

=(

2n3 + (β2 − 4β + 6)n2 + (12β − 5β2)n + 6β2

(n − 2)(n − 3)(n + β)2

)φ2(x)

+(

(6 + 4α − 2β − 2αβ)n2 + (6β + 10αβ − 12α)n − 12αβ − (β2 − 4β + 6)n2 − (12β − 5β2)n − 6β2

(n − 2)(n − 3)(n + β)2

)x

+(2 + α2 + 2α)n2 − (6α + 5α2)n + 6α2

(n − 2)(n − 3)(n + β)2

≤(

2n3 + (β2 − 4β + 6)n2 + (12β − 5β2)n + 6β2

(n − 2)(n − 3)(n + β)2

)φ2(x) +

(2 + α2 + 2α)n2 − (6α + 5α2)n + 6α2

(n − 2)(n − 3)(n + β)2.

Using α ≤ β and n < n2 < n3 for n > 3, we have

D(α,β)n ((t − x)2, x) ≤

((2 + β2 − 4β + 6 + 12β − 5β2 + 6β2)n3

(n − 2)(n − 3)(n + β)2

)φ2(x)

+(2 + α2 + 2α − 6α − 5α2 + 6α2)n2

(n − 2)(n − 3)(n + β)2

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APPROXIMATION BY THE DURRMEYER-BASKAKOV-STANCU OPERATORS 275

≤(

(7β2 + 12β + 8)n3

(n − 2)(n − 3)(n + β)2

)φ2(x) +

(7α2 + 2α + 2)n2

(n − 2)(n − 3)(n + β)2

≤ 7β2 + 12β + 8(n − 2)

(φ2(x) +

1n − 3

),

which completes the proof. �

3. LOCAL APPROXIMATIONLet the space CB [0,∞) of all continuous and bounded functions be endowed with the norm ||f || =

sup{|f(x)| : x ∈ [0,∞)}. Further let us consider the following K-functional:

K2(f, δ) = infg∈W 2

{||f − g|| + δ||g′′||}, (3.1)

where δ > 0 and W 2 = {g ∈ CB [0,∞) : g′, g′′ ∈ CB [0,∞)}. By the method as given ([6] p. 177 Theo-rem 2.4), there exists an absolute constant C > 0 such that

K2(f, δ) ≤ Cω2(f,√

δ), (3.2)

where

ω2(f,√

δ) = sup0<h≤

√δ

supx∈[0,∞)

|f(x + 2h) − 2f(x + h) + f(x)| (3.3)

is the second order modulus of smoothness of f ∈ CB[0,∞). Also we set

ω(f,√

δ) = sup0<h≤

√δ

supx∈[0,∞)

|f(x + h) − f(x)|. (3.4)

We denote the usual modulus of continuity of f ∈ CB [0,∞). In what follows we shall use the notationsφ(x) =

√x(x + 1) and δ2

n(x) = φ2(x) + 1n−3 , where x ∈ [0,∞) and n ≥ 4.

Our first result is a direct local approximation theorem for the operators D(α,β)n .

Theorem 1. Let f ∈ CB[0,∞). Then, we have following inequality,

|D(α,β)n (f, x) − f(x)| ≤ Cω2

(f,

√20β + 11β2 + 12

(n − 2)δn(x)

)

+ ω

(f,

(2n − nβ + 2β)x + n + nα − 2α(n − 2)(n + β)

),

where C is a positive constant.

Proof. Let us define the auxiliary operator L(α,β)n by

L(α,β)n (f, x) = D(α,β)

n (f, x) + f(x) − f

(x +

(2n − nβ + 2β)x + n + nα − 2α(n − 2)(n + β)

), (3.5)

for every x ∈ [0,∞). The operator L(α,β)n are linear and preserve the linear functions:

L(α,β)n (t − x, x) = 0, t ∈ [0,∞). (3.6)

Let g ∈ W 2 and x, t ∈ [0,∞). By Taylor’s expansion

g(t) = g(x) + g′(x)(t − x) +

t∫

x

(t − u)g′′(u)du, t ∈ [0,∞).

Applying L(α,β)n and by (3.6), we get

L(α,β)n (g, x) = g(x) + L(α,β)

n

⎛

⎝t∫

x

(t − u)g′′(u)du, x

⎞

⎠ .

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276 MISHRA, PATEL

Hence by (3.5) one has

|L(α,β)n (g, x) − g(x)| ≤

∣∣∣∣∣∣D(α,β)

n

⎛

⎝t∫

x

(t − u)g′′(u)du, x

⎞

⎠

∣∣∣∣∣∣

+

∣∣∣∣∣∣∣∣

x+(2n−nβ+2β)x+n+nα−2α

(n−2)(n+β)∫

x

(x +

(2n − nβ + 2β)x + n + nα − 2α(n − 2)(n + β)

− u

)g′′(u)du

∣∣∣∣∣∣∣∣

≤ D(α,β)n ((t − x)2||g′′||, x) +

((2n − nβ + 2β)x + n + nα − 2α

(n − 2)(n + β)

)2

||g′′||,

|L(α,β)n (g, x) − g(x)| ≤

(D(α,β)

n ((t − x)2, x) +(

(2n − nβ + 2β)x + n + nα − 2α(n − 2)(n + β)

)2)||g′′||. (3.7)

Now, Using α ≤ β and n < n2 < n3 for n > 3,(

(2n − nβ + 2β)x + n + nα − 2α(n − 2)(n + β)

)2

=(2n − nβ + 2β)2x2 + 2(2n − nβ + 2β)(n + nα − 2α)x + (n + nα − 2α)2

(n − 2)2(n + β)2

=(2n − nβ + 2β)2φ2(x) + {2(2n − nβ + 2β)(n + nα − 2α) − (2n − nβ + 2β)2}x

(n − 2)2(n + β)2

+(n + nα − 2α)2

(n − 2)2(n + β)2≤ (2n − nβ + 2β)2φ2(x) + (n + nα − 2α)2

(n − 2)2(n + β)2

≤ 4(β + 1)2

n − 2

(φ2(x) +

1n − 3

).

There fore by (3.7) and Lemma 3, we have

|L(α,β)n (g, x) − g(x)| ≤

[7β2 + 12β + 8

n − 2δ2n(x) +

4(β2 + 2β + 1)n − 2

δ2n(x)

]||g′′||,

|L(α,β)n (g, x) − g(x)| ≤ 12 + 20β + 11β2

n − 2δ2n(x)||g′′||. (3.8)

On the other hand, by (3.5), (3.8) and by Lemma 4, we get

|L(α,β)n (f, x)| ≤ |D(α,β)

n (f, x)| + 2||f || ≤ ||f ||D(α,β)n (1, x) + 2||f || ≤ 3||f ||. (3.9)

Now by (3.8), (3.9) and (3.5), imply

|D(α,β)n (f, x) − f(x)| ≤ |L(α,β)

n (f − g, x) − (f − g)(x)| + |L(α,β)n (g, x) − g(x)|

+∣∣∣∣f

(x +

(2n − nβ + 2β)x + n + nα − 2α(n − 2)(n + β)

)− f(x)

∣∣∣∣

≤ 4||f − g|| + 12 + 20β + 11β2

n − 2δ2n(x)||g′′|| +

∣∣∣∣f(

x +(2n − nβ + 2β)x + n + nα − 2α

(n − 2)(n + β)

)− f(x)

∣∣∣∣.

Hence taking infimum on the right hand side over all g ∈ W 2, we get

|D(α,β)n (f, x) − f(x)| ≤ K2

(f,

12 + 20β + 11β2

n − 2δ2n(x)

)+ ω

(f,

(2n − nβ + 2β)x + n + nα − 2α(n − 2)(n + β)

).

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APPROXIMATION BY THE DURRMEYER-BASKAKOV-STANCU OPERATORS 277

In view of (3.2),

|D(α,β)n (f, x) − f(x)| ≤ Cω2

(f,

√12 + 20β + 11β2

n − 2δn(x)

)

+ ω

(f,

(2n − nβ + 2β)x + n + nα − 2α(n − 2)(n + β)

),

which prove the theorem. �

4. RATE OF CONVERGENCE

Let Bx2 [0,∞) be the set of functions define on [0,∞) stratifying the condition |f(x)| ≤ Mf (1 + x2),where Mf is a constant depending on only f . By Cx2[0,∞), we denote subspace of all continuousfunctions belonging to Bx2[0,∞). Also,let C∗

x2 [0,∞) be the subspace of all f ∈ Cx2 [0,∞) for which

limx→∞f(x)1+x2 is finite. The norm on C∗

x2[0,∞) if ||f ||x2 = supx∈[0,∞)|f(x)|1+x2 . For any positive number a,

by

ωa(f, δ) = sup|t−x|≥δ

supx,t∈[0,a]

|f(t) − f(x)|,

we denote the usual modulus of continuity of f on the closed interval [0, a]. We know that for a functionf ∈ Cx2[0,∞), the modulus of continuity ωa(f, δ) tends to zero. Now we give a rate of convergence

theorem for the operator D(α,β)n .

Theorem 2. Let f ∈ Cx2 [0,∞)and ωa+1(f, δ) be its modulus of continuity on the finite interval[0, a + 1] ⊂ [0,∞), where a > 0 then for every n > 3

||D(α,β)n (f) − f ||C[0,a] ≤

K

n − 2

(φ2(x) +

1n − 3

)

+ 2ωa+1

(f,

√12 + 20β + 11β2

n − 2

(φ2(x) +

1n − 3

)),

where K = 6Mf (1 + a2)(12 + 20β + 11β2).Proof. For x ∈ [0, a] and t > a + 1. Since t − x > 1, we have

|f(x) − f(t)| ≤ Mf (2 + x2)(t − x)2 ≤ Mf (2 + 3x2 + 2(t − x)2)

≤ Mf (2 + x2)(t − x)2 ≤ 3Mf (1 + x2 + (t − x)2),

|f(x) − f(t)| ≤ 6Mf (1 + a2)(t − x)2. (4.1)

For x ∈ [0, a] and t ≤ a + 1, we have

|f(t) − f(x)| ≤ ωa+1(f, |t − x|) ≤(

1 +|t − x|

δ

)ωa+1(f, δ), (4.2)

with δ > 0. From (4.1) and (4.2), we get

|f(t) − f(x)| ≤ 6Mf (1 + a2)(t − x)2 +(

1 +|t − x|

δ

)ωa+1(f, δ).

For x ∈ [0, a] and t ≥ 0

|D(α,β)n (f, x) − f(x)| ≤ D(α,β)

n (|f(x) − f(t)|, x)

≤ 6Mf (1 + a2)D(α,β)n ((t − x)2, x) + ωa+1(f, δ)

(1 +

1δ[D(α,β)

n ((t − x)2, x)]12

).

Hence, by Schwarz’s inequality and lemma 3 for every x ∈ [0, a],

|D(α,β)n (f, x) − f(x)| ≤ 6Mf (1 + a2)(12 + 20β + 11β2)

n − 2

(φ2(x) +

1n − 3

)

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278 MISHRA, PATEL

+ ωa+1(f, δ)

(1 +

1δ

√12 + 20β + 11β2

n − 2

(φ2(x) +

1n − 3

)),

by taking δ =√

12+20β+11β2

n−2

(φ2(x) + 1

n−3

), we get the assertion our theorem

||D(α,β)n (f) − f ||C[0,a] ≤

K

n − 2

(φ2(x) +

1n − 3

)

+ 2ωa+1

(f,

√12 + 20β + 11β2

n − 2

(φ2(x) +

1n − 3

)).

�

Corollary 1. If f ∈ LipM (α) on [0, a + 1], then for n > 3

||D(α,β)n (f) − f || ≤ (1 + 2M)

√K

n − 3.

Proof. For Sufficient large n,

K

n − 3≤

√K

n − 3,

becuase of limn→∞(n − 3) = ∞. Hence,by f ∈ LipMα, we abtain the assertion of the corollary. �

5. WEIGHTED APPROXIMATION

Now we shall discuss the weighted approximation theorem, when the approximation formula holdstrue on the interval [0,∞).

Theorem 3. For each f ∈ C∗x2[0,∞), we have

limn→∞

||D(α,β)n (f) − f ||x2 = 0.

Proof. Using the theorem in [7] we see that it is sufficient to verify the following three conditions

limn→∞

||D(α,β)n (tr, x) − xr||x2 = 0, r = 0, 1, 2. (5.1)

Since, D(α,β)n (1, x) = 1, the first condition of (5.1) is satisfied for r = 0. Now,

||D(α,β)n (t, x) − x||x2 = sup

x∈[0,∞)

|D(α,β)n (t, x) − x|

1 + x2≤ sup

x∈[0,∞)

∣∣∣∣n2x + (1 + α)n − 2α

(n + β)(n − 2)− x

∣∣∣∣

× 11 + x2

≤∣∣∣∣n2 − (n + β)(n − 2)

(n + β)(n − 2)

∣∣∣∣ supx∈[0,∞)

x

1 + x2+

(1 + α)n − 2α(n + β)(n − 2)

≤ 2n − nβn − 2β(n + β)(n − 2)

+(1 + α)n − 2α(n + β)(n − 2)

→ 0 as n → ∞.

There for, condition of (5.1) hold for r = 1. Similarly, we can write for n > 3

||D(α,β)n (t2, x) − x2||x2 = sup

x∈[0,∞)

|D(α,β)n (t2, x) − x2|

1 + x2

≤(

(n + 1)n3

(n − 2)(n − 3)(n + β)2− 1

)sup

x∈[0,∞)

x2

1 + x2

+n2(4n + 2α(n − 3))

(n − 2)(n − 3)(n + β)2sup

x∈[0,∞)

x

1 + x2+

(2 + 2α + 2α2)n2 − (6α + 5α2)n2 + 6α2

(n − 2)(n − 3)(n + β)2

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APPROXIMATION BY THE DURRMEYER-BASKAKOV-STANCU OPERATORS 279

≤ 2n3 + (β2 − 4β + 6)n2 + (12β − 5β2)n + 6β2

(n − 2)(n − 3)(n + β)2

+2n3 + (6 + 4α − 2β − 2αβ)n2 + (6β + 10αβ − 12α)n − 12αβ

(n − 2)(n − 3)(n + β)2

+(2 + α2 + 2α)n2 − (6α + 5α2)n + 6α2

(n − 2)(n − 3)(n + β)2,

which implies that ||D(α,β)n (t2, x) − x2||x2 → 0 as n → ∞. Thus proof is completed. �

We give the following theorem to approximate all functions in Cx2[0,∞). This type of results are givenin [8] for locally integrable functions.

Theorem 4. For each f ∈ Cx2[0,∞) and α > 0, we have limn→∞ supx∈[0,∞)|D(α,β)

n (f,x)−f(x)|(1+x2)1+α = 0.

Proof. For any fixed x0 > 0,

supx∈[0,∞)

|D(α,β)n (f, x) − f(x)|

(1 + x2)1+α≤ sup

x≤x0

|D(α,β)n (f, x) − f(x)|

(1 + x2)1+α+ sup

x≥x0

|D(α,β)n (f, x) − f(x)|

(1 + x2)1+α

≤ ||D(α,β)n (f) − f ||C[0,x0] + ||f ||x2 sup

x≥x0

|D(α,β)n (1 + t2, x)|(1 + x2)1+α

+ supx≥x0

|f(x)|(1 + x2)1+α

.

The first term of the above inequality tends to zero from Theorem 2. By Lemma 3 for any fixed x0 > 0 it

is easily seen that supx≥x0

|D(α,β)n (1+t2,x)|(1+x2)1+α tends to zero as n → ∞. We can choose x0 > 0 so large that

the last part of the above inequality can be made small enough. Thus the proof is completed. �

6. VORONOVSKAJA TYPE THEOREM

In this section we establish a Voronovskaja type asymptotic formula for the operators D(α,β)n :

Lemma 4. For every x ∈ [0,∞), we have

limn→∞

nD(α,β)n (t − x, x) = (2 − β)x + 1 + α, (6.1)

limn→∞

nD(α,β)n ((t − x)2, x) = 2x(1 + x). (6.2)

Theorem 5. If any f ∈ C∗x2[0,∞) such that f ′, f ′′ ∈ C∗

x2[0,∞) and x ∈ [0,∞) then, we have

limn→∞

n(D(α,β)n (f, x) − f(x)) = (α + 1 + (2 − β)x)f ′(x) + x(1 + x)f ′′(x).

Proof. Let f, f ′, f ′′ ∈ C∗x2[0,∞) and x ∈ [0,∞) be fixed. By Taylor’s expansion we can write

f(t) = f(x) + f ′(x)(t − x) +12f ′′(x)(t − x)2 + r(x, t)(t − x)2, (6.3)

where r(t, x) is Peano form of the remainder, r(·, x) ∈ C∗x2 [0,∞) and limt→x r(t, x) = 0. ApplyingD(α,β)

n

to above, we obtain

n[D(α,β)n (f, x) − f(x)] = f ′(x)nD(α,β)

n (t − x, x)

+n

2f ′′(x)D(α,β)

n ((t − x)2, x) + nD(α,β)n (r(t, x)(t − x)2, x).

By Cauchy-Schwarz inequality, we have

D(α,β)n (r(t, x)(t − x)2, x) ≤

√D(α,β)

n (r(t, x)2, x)√

D(α,β)n ((t − x)4, x). (6.4)

Observer that r2(x, x) = 0 and r2(·, x) ∈ C∗x2[0,∞). Then it follows from Theorem that

limn→∞

nD(α,β)n (r(t, x)2, x) = r2(x, x) = 0, (6.5)

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 34 No. 3 2013

280 MISHRA, PATEL

uniformly with respect to x ∈ [0, A]. Now from (6.4) and (6.5) and Lemma 4, we obtain

limn→∞

nD(α,β)n (r(t, x)(t − x)2, x) = 0.

Notice that,

limn→∞

n[D(α,β)n (f, x) − f(x)]

= limn→∞

(f ′(x)nD(α,β)

n (t − x, x) +n

2f ′′(x)D(α,β)

n ((t − x)2, x) + nD(α,β)n (r(t, x)(t − x)2, x)

)

= ((2 − β)x + 1 + α)f ′(x) + (x2 + x)f ′′(x).

�

7. BETTER ESTIMATION

In this section we give better estimates for the operators D(α,β)n as follows:

Many well know operators preserves the linear as well as constant function for example Bernstein,Baskakov, Szasz-Mirakyan and Szasz-beta operators posses these properties that means Ln(ei, x) =ei(x) where ei(x) = xi(i = 0, 1). To make the convergence faster King [9] proposed an approach tomodify the classical Bernstein polynomial, so that the sequence preserve test function e0 and e2. After

that this approach was applied to some well known operators. As the Operator D(α,β)n preserve the

constant as well as linear functions, for this purpose the modification of D(α,β)n as follows:

D∗(α,β)n (f, x) = (n − 1)

∞∑

k=0

pn,k(rn(x))

∞/A∫

0

pn,k(t)f(

nt + α

n + β

)dt, (7.1)

where rn(x) = (n−2)(n+β)x−n−αn−2n2 and x ∈ In =

[n+α(n−2)

(n−2)(n+β) ,∞)

.

Lemma 5. For each x ∈ In, we have

D∗(α,β)n (1, x) = 1, D∗(α,β)

n (t, x) = x,

D∗(α,β)n (t2, x) =

(n + 1)(n − 2)n(n − 3)

x2 − 2(n + 1)[n + α(n − 2)]n(n − 3)

x

+4n + 2α(n − 3)(n − 3)(n + β)

x +(n + α(n − 2))2(n + 1)n(n − 2)(n − 3)(n + β)2

− (n + α)(n − 2)(4n + 2α(n − 3)(n − 2)(n − 3)(n + β)2

+(2 + 2α + α2)n2 − (6α + 5α2)n + 6α2

(n − 2)(n − 3)(n + β)2.

Remark 3. Form Above Lemma and for n > 3 by simple computation, we have

D∗(α,β)n ((t − x)2, x) ≤ 11β2 + 20β + 12

n − 2

(φ2(x) +

1n − 3

)= ρn(x)(say),

where φ(x) =√

x(1 + x), x ∈ In.Theorem 6. Let f ∈ CB(In). Then for all x ∈ In and n > 3, there exist an absolute constant

M > 0 such that

|D∗(α,β)n (f, x) − f(x)| ≤ Mω2(f,

√ρn).

Proof. Let g ∈ CB(In) and x, t ∈ In. Using, the Taylors expansion, we have

g(t) − g(x) = (t − x)g′(x) +

t∫

x

(t − u)g′′(u)du.

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 34 No. 3 2013

APPROXIMATION BY THE DURRMEYER-BASKAKOV-STANCU OPERATORS 281

Applying D∗(α,β)n both side,

D∗(α,β)n (g, x) − g(x) = g′(x)D∗(α,β)

n (t − x, x) + D∗(α,β)n

⎛

⎝t∫

x

(t − u)g′′(u)du, x

⎞

⎠ .

Obviously, we have |∫ tx(t − u)g′′(u)du| ≤ (t − x)2||g′′|| and D∗(α,β)

n (t − x, x) = 0. There fore,

|D∗(α,β)n (g, x) − g(x)| ≤ ||g′′||D∗(α,β)

n ((t − x)2, x) ≤ ρn(x)||g′′||.

Since, |D∗(α,β)n (f, x)| ≤ ||f ||

|D∗(α,β)n (f, x) − f(x)| ≤ |D∗(α,β)

n (f − g, x) − (f − g)(x)| + |D∗(α,β)n (g, x)|

≤ 2||f − g|| + ρn(x)||g′′||.

Taking infimum over all g ∈ C2(In), we obtain

|D∗(α,β)n (f, x) − f(x)| ≤ K2(f, ρn).

In view of (3.2), we have

|D∗(α,β)n (f, x) − f(x)| ≤ Cω2(f,

√ρn),

which prove the theorem. �

Theorem 7. If f ∈ C∗x2(In) such that f ′, f ′′ ∈ C∗

x2(In). Then, we have

limn→∞

n[D∗(α,β)n (f, x) − f(x)] = x(1 + x)f ′′(x).

The proof follows from along the lines of Theorem 5.

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10. Z. Finta and V. Gupta, South East Asian Bull. Math. 6, 1037 (2008).11. N. K. Govil and V. Gupta, Nonlinear Anal. 69 (11), 3795 (2008).12. I. Yksel and N. Ispir, Comput. Math. Appl. 52, 1463 (2006).13. G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its

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