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Open Access Original research

Convergence in simultaneous approximation for Srivastava-Gupta operators

Durvesh Kumar Verma* and Purshottam N Agrawal

Author Affiliations

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

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Mathematical Sciences 2012, 6:22  doi:10.1186/2251-7456-6-22


The electronic version of this article is the complete one and can be found online at: http://www.iaumath.com/content/6/1/22


Received:18 May 2012
Accepted:12 July 2012
Published:16 August 2012

© 2012 Verma and Agrawal; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Purpose

The purpose of the present paper is to introduce the generalized form of Srivastava-Gupta operators and study their approximation properties.

Methods

We use analytical method to obtain our results.

Results

We have established the rate of convergence, in simultaneous approximation, for functions having derivatives of bounded variation.

Conclusions

The results proposed here are new and have a better rate of convergence.

Keywords:
Bounded variation; Srivastava-Gupta operators; Simultaneous approximation; Rate of convergence; 2010; 26A45; 41A28

Introduction

In the year 2003, Srivastava and Gupta [1] introduced a general family of summation-integral type operators which includes some well-known operators as special cases. They estimated the rate of convergence for functions of bounded variation. For the details of special cases in [1], we refer the readers to [2-7]. Ispir and Yuksel [8] considered the Bezier variant of the operators studied in [1] and estimated the rate of convergence for functions of bounded variation. Very recently, Deo [9] studied Srivastava-Gupta operators and obtained the faster rate convergence as well as Voronovskaja type results for these operators by using the King approach. In the last section, he considered Stancu variant of these operators and established some approximation properties.

The operators Gn,c is defined as follows:

G n , c ( f , x ) = n k = 1 p n , k ( x , c ) 0 p n + c , k 1 ( t , c ) f ( t ) dt + p n , 0 ( x , c ) f ( 0 ) , (1.1)

where

p n , k ( x , c ) = ( x ) k k ! ϕ n , c ( k ) ( x ) (1.2)

and

ϕ n , c ( x ) = e nx , c = 0 , ( 1 + cx ) n / c , c = 1 , 2 , 3 , ..

Here { ϕ n , c ( x ) } n = 1 is a sequence of functions defined on the closed interval [0,b, b>0, satisfying the following properties. For each n N and k N 0 : = N { 0 } :

(i) ϕ n , c C [ a , b ] , b > a 0 ,

(ii) ϕn,c(0)=1,

(iii) ϕn,c is completely monotone so that ( 1 ) k ϕ n , c ( k ) ( x ) 0 , x [ 0 , b ] , and

(iv) there exists an integer c such that

ϕ n , c ( k + 1 ) ( x ) = n ϕ n + c , c ( k ) ( x ) , n > max { 0 , c } ; x [ 0 , b ]

(see [1]).

Nowadays, the rate of convergence for the functions having the derivatives of bounded variation (BV) is an interesting area of research. Bai et al.[10] worked in this direction and estimated the rate of convergence for the several operators. Gupta [4] estimated the rate of convergence for functions of BV on certain Baskakov-Durrmeyer type operators. Ispir et al. [11] estimated the rate of convergence for the Kantorovich type operators for functions having derivatives of BV. Recently, Acar et al. [12] introduced the general integral modification of the Szász-Mirakyan operators having the weight functions of Baskakov basis functions. The rate of convergence for functions having the derivatives of bounded variation is obtained. This motivated us to study the rate of convergence for the generalized Srivastava-Gupta operators as follows: For a function f B V α [ 0 , ) , the class of bounded variation functions satisfying the growth condition |f (t )|≤M (1 + t)αM>0, α≥0, the operators Gn,r,c are defined by

G n , r , c ( f , x ) = n Γ ( n c + r ) Γ ( n c r + 1 ) Γ ( n c + 1 ) Γ ( n c ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) f ( t ) dt , (1.3)

where pn,k(xc) is given by Equation 1.2 and n>(r−1)c.

Remark 1

For the special case of c=1, the operators in Equation 1.3 are reduced to the following operators:

G n , r , 1 ( f , x ) = ( n + r 1 ) ! ( n r ) ! ( ( n 1 ) ! ) 2 k = 0 p n + r , k ( x , 1 ) 0 p n ( r 1 ) , k + r 1 ( t , 1 ) f ( t ) dt ,

where p n , k ( x , 1 ) = n + k 1 k x k ( 1 + x ) n + k .

We denote that the class of absolutely continuous functions f on (0,) by DBq(0,), (where q is some positive integer) are satisfied:

(i) | f ( t ) | C 1 t q , C 1 > 0 and

(ii) the function f has the first derivative on interval (0,) which coincide, a.e., with a function which is of bounded variation on every finite subinterval of (0,). It can be observed that for all fDBq(0,), we can have the representation

f ( x ) = f ( c ) + c x ψ ( t ) dt , x c > 0 .

In the present paper, we study the rate of convergence for the operators Gn,r,c for functions having the derivatives of bounded variation. We also mention a corollary which provides the result in simultaneous approximation.

Methods

The principal methods used in the present work involve the application of the theory of functions having the derivatives of bounded variation to analyze and study the rate of convergence, in simultaneous approximation, for the Srivastava-Gupta operators.

Results and discussion

In the sequel we shall need the following lemmas:

Lemma 1

If we define the moments as

T n , r , m ( x , c ) = ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) ( t x ) m dt ,

and then, Tr,n,0(x,c)=1, T r , n , 1 ( x , c ) = ( 1 + 2 r ) cx + r n ( r + 1 ) c and for n>(m + r + 1)c, we have the following recurrence relation:[n−(m + r + 1)c]Tn,r,m + 1(x,c)

= x ( 1 + cx ) [ T n , r , m ( x , c ) + 2 m T n , r , m 1 ( x , c ) ] + [ ( m + r ) ( 1 + 2 cx ) + cx ] T n , r , m ( x , c ) .

Consequently,

T n , r , 2 ( x , c ) = x ( 1 + cx ) ( 2 n c ) + [ ( 1 + r ) ( 1 + 2 cx ) + cx ] · [ ( 1 + 2 r ) cx + r ] ( n ( r + 1 ) c ) ( n ( r + 2 ) c ) .

Furthermore, Tn,r,m(x,c) is polynomial of degree m in x and

T n , r , m ( x , c ) = O n m + 1 2 .

Proof

Taking the derivative of Tn,r,m(x,c) with respect to x and using the identity x ( 1 + cx ) p n + rc , k ( x , c ) = [ k ( n + rc ) x ] p n + rc , k ( x , c ) , we have

x ( 1 + cx ) [ T n , r , m ( x , c ) + m T n , r , m 1 ( x , c ) ] = ( n rc ) k = 0 [ k ( n + rc ) x ] p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) t x m dt = ( n rc ) k = 0 p n + rc , k ( x , c ) 0 [ ( k + r 1 ) ( n ( r 1 ) c ) t ] p n ( r 1 ) c , k + r 1 ( t , c ) t x m dt + [ n ( r 1 ) c ] ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) t t x m dt [ ( n + rc ) x + ( r 1 ) ] T n , r , m ( x , c ) = I 1 + I 2 [ ( n + rc ) x + ( r 1 ) ] T n , r , m ( x , c ) .

To compute I2 we have

I 2 = [ n ( r 1 ) c ] ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) × t x m + 1 + x t x m dt = [ n ( r 1 ) c ] T n , r , m + 1 ( x , c ) + x T n , r , m ( x , c ) .

Using t ( 1 + ct ) p n ( r 1 ) c , k ( t , c ) = [ k ( n ( r 1 ) c ) t ] pn−(r−1)c,k(t,c), we can write I1 as

I 1 = ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) t t x m dt + ( n rc ) c k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) t 2 t x m dt = J 1 + J 2 .

Again using t(tx)m=(tx)m + 1 + x(tx)m and integrating by parts, we get

J 1 = ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) t x m + 1 + x t x m dt = ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) × ( m + 1 ) t x m mx t x m 1 dt = ( m + 1 ) T n , r , m ( x , c ) mx T n , r , m 1 ( x , c ) .

Proceeding in a similar manner, we obtain J2 as

J 2 = c ( m + 2 ) T n , r , m + 1 ( x , c ) 2 ( m + 1 ) x T n , r , m ( x , c ) m x 2 T n , r , m 1 ( x , c ) .

Combining I1,I2,J1, and J2, we have

x ( 1 + cx ) [ T n , r , m ( x , c ) + m T n , r , m 1 ( x , c ) ] = ( m + 1 ) T n , r , m ( x , c ) mx T n , r , m 1 ( x , c ) + c ( m + 2 ) T n , r , m + 1 ( x , c ) 2 ( m + 1 ) x T n , r , m ( x , c ) m x 2 T n , r , m 1 ( x , c ) + [ n ( r 1 ) c ] T n , r , m + 1 ( x , c ) + x T n , r , m ( x , c ) [ ( n + rc ) x + ( r 1 ) ] T n , r , m ( x , c ) ,

x ( 1 + cx ) [ T n , r , m ( x , c ) + m T n , r , m 1 ( x , c ) ] = { n ( r 1 ) c } ( m + 2 ) c T n , r , m + 1 ( x , c ) + ( m + 1 ) 2 ( m + 1 ) cx + { n ( r 1 ) c } x { ( n + rc ) x + ( r 1 ) } T n , r , m ( x , c ) + mx mc x 2 T n , r , m 1 ( x , c ) , and

[ n ( m + r + 1 ) c ] T n , r , m + 1 ( x , c ) = x ( 1 + cx ) [ T n , r , m ( x , c ) + 2 m T n , r , m 1 ( x , c ) ] + [ ( m + r ) ( 1 + 2 cx ) + cx ] T n , r , m ( x , c ) .

Remark 2

Let x∈(0,) and λ>2; then for n sufficiently large, Lemma 1 yields that

T n , r , 2 ( x , c ) λx ( 1 + cx ) n ( c N 0 ) .

Lemma 2

Let x∈(0,) and λ>2; then for n sufficiently large, we have

μ n , r ( x , y ) = ( n rc ) k = 0 p n + rc , k ( x , c ) 0 y p n ( r 1 ) c , k + r 1 ( t , c ) dt λx ( 1 + cx ) n ( x y ) 2 , 0 y < x , 1 μ n , r ( x , z ) = ( n rc ) k = 0 p n + rc , k ( x , c ) z p n ( r 1 ) c , k + r 1 ( t , c ) dt λx ( 1 + cx ) n ( z x ) 2 , x < z < ∞.

Proof

The proof of the lemma follows easily by Remark 2. For instance, for the first inequality for n sufficiently large and 0≤y<x, we have

μ n , r ( x , y ) = ( n rc ) k = 0 p n + rc , k ( x , c ) 0 y p n ( r 1 ) c , k + r 1 ( t , c ) dt ( n rc ) k = 0 p n + rc , k ( x , c ) 0 y p n ( r 1 ) c , k + r 1 ( t , c ) ( t x ) 2 ( y x ) 2 dt T n , r , 2 ( x ) ( y x ) 2 λx ( 1 + cx ) n ( y x ) 2 .

The proof of the second inequality follows along the similar lines. □

Lemma 3

Let f be s times differentiable on [0,) such that f(s−1)(t)=O(tq) as t where q is a positive integer. Then for any r , s N 0 and n>max{q,r + s + 1}, we have

D s G n , r , c ( f , x ) = G n , r + s , c ( D s f , x ) , D = d dx .

Proof

We prove this result by applying the principle of mathematical induction and using the following identity:

p n , k ( x , c ) = n [ p n + c , k 1 ( x , c ) p n + c , k ( x , c ) ] and p n , 0 ( x , c ) = n p n + c , 0 ( x , c ) . (2.1)

The above identities are true even for the case of k=0, as we observe that pn + c,k=0 for k<0. Using Equation 2.1 and integrating by parts, we have

D [ G n , r , c ] ( f , x ) = n Γ ( n c + r ) Γ ( n c r + 1 ) Γ ( n c + 1 ) Γ ( n c ) k = 0 D p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) f ( t ) dt

= n Γ ( n c + r ) Γ ( n c r + 1 ) Γ ( n c + 1 ) Γ ( n c ) · k = 0 ( n + rc ) [ p n + ( r + 1 ) c , k 1 ( x , c ) p n + ( r + 1 ) c , k ( x , c ) ] 0 p n ( r 1 ) c , k + r 1 ( t , c ) f ( t ) dt = n ( n + rc ) Γ ( n c + r ) Γ ( n c r + 1 ) Γ ( n c + 1 ) Γ ( n c ) · k = 0 p n + ( r + 1 ) c , k ( x , c ) 0 [ p n ( r 1 ) c , k + r ( t , c ) p n ( r 1 ) c , k + r 1 ( t , c ) ] f ( t ) dt = n ( n + rc ) Γ ( n c + r ) Γ ( n c r + 1 ) ( n rc ) Γ ( n c + 1 ) Γ ( n c ) · k = 0 p n + ( r + 1 ) c , k ( x , c ) 0 D p n rc , k + r ( t , c ) f ( t ) dt = n Γ ( n c + r + 1 ) Γ ( n c r ) Γ ( n c + 1 ) Γ ( n c ) · k = 0 p n + ( r + 1 ) c , k ( x , c ) 0 p n rc , k + r ( t , c ) Df ( t ) , dt = [ G n , r + 1 , c ] ( Df , x ) ,

which shows that the result holds for s=1. Let us suppose that the result holds for s=m i.e.,

D m [ G n , r , c ] ( f , x ) = [ G n , r + m , c ] ( D m f , x ) = n Γ ( n c + r + m ) Γ ( n c r m + 1 ) Γ ( n c + 1 ) Γ ( n c ) k = 0 p n + ( r + m ) c , k ( x , c ) 0 p n ( r + m 1 ) c , k + r + m 1 ( t , c ) D m f ( t ) dt.

Now by Equation 2.1,

D m + 1 [ G n , r , c ] ( f , x )

= n Γ ( n c + r + m ) Γ ( n c r m + 1 ) Γ ( n c + 1 ) Γ ( n c ) k = 0 D p n + ( r + m ) c , k ( x , c ) 0 p n ( r + m 1 ) c , k + r + m 1 ( t , c ) D m f ( t ) dt

= n Γ ( n c + r + m ) Γ ( n c r m + 1 ) Γ ( n c + 1 ) Γ ( n c ) k = 0 [ n + ( r + m ) c ] × p n + ( r + m + 1 ) c , k 1 ( x , c ) p n + ( r + m + 1 ) c , k ( x , c ) 0 p n ( r + m 1 ) c , k + r + m 1 ( t , c ) D m f ( t ) dt = nc Γ ( n c + r + m + 1 ) Γ ( n c r m + 1 ) Γ ( n c + 1 ) Γ ( n c ) k = 0 p n + ( r + m + 1 ) c , k 0 p n ( r + m 1 ) c , k + r + m ( t , c ) p n ( r + m 1 ) c , k + r + m 1 ( t , c ) D m f ( t ) dt = nc Γ ( n c + r + m + 1 ) Γ ( n c r m + 1 ) Γ ( n c + 1 ) Γ ( n c ) k = 0 p n + ( r + m + 1 ) c , k 0 D p n ( r + m ) c , k + r + m ( t , c ) n ( r + m 1 ) c D m f ( t ) dt.

Integrating by parts the last integral, we have

D m + 1 [ G n , r , c ] ( f , x ) = n Γ ( n c + r + m + 1 ) Γ ( n c r m ) Γ ( n c + 1 ) Γ ( n c ) k = 0 p n + ( r + m + 1 ) c , k 0 p n ( r + m ) c , k + r + m ( t , c ) D m + 1 f ( t ) dt.

Therefore,

D m + 1 G n , r , c ( f , x ) = G n , r + m + 1 , c ( D m + 1 f , x ) .

Thus, the result is true for s=m + 1; hence, by mathematical induction the proof of the lemma is completed. □

Main results

In this subsection we prove our main results.

Theorem 1

Let fDBq(0,), q>0 and x∈(0,). The for λ>2 and n sufficiently large, we have ( Γ ( n c ) ) 2 Γ ( n c + r ) Γ ( n c r ) G n , r , c ( f ; x ) f ( x )

λ ( 1 + cx ) n k = 1 [ n ] x x / k x + x / k ( ( f ) x ) + x n x x / n x + x / n ( ( f ) x ) + λ ( 1 + cx ) n ( | f ( 2 x ) f ( x ) x f ( x + ) | + | f ( x ) | ) + O ( n q ) + λ ( 1 + cx ) n | f ( x + ) | + | f ( x + ) f ( x ) | 2 λx ( 1 + cx ) n + | f ( x + ) + f ( x ) | 2 r ( 1 + 2 cx ) + cx n ( r + 1 ) c ,

where a b f x denotes the total variation of fx on [a,b], and the auxiliary function fx is defined by

f x ( t ) = f ( t ) f ( x ) , 0 t < x , 0 , t = x , f ( t ) f ( x + ) , x < t < ∞.

Proof

Using the mean value theorem, we have

( Γ ( n c ) ) 2 Γ ( n c + r ) Γ ( n c r ) G n , r , c ( f ; x ) f ( x ) = ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) × [ f ( t ) f ( x ) ] dt = 0 x t ( n rc ) k = 0 p n + rc , k ( x , c ) p n ( r 1 ) c , k + r 1 × ( t , c ) f ( u ) du dt.

Also it is a valid identity that

f ( u ) = f ( x + ) + f ( x ) 2 + ( f ) x ( u ) + f ( x + ) f ( x ) 2 sgn ( u x ) + f ( x ) f ( x + ) + f ( x ) 2 χ x ( u ) ,

where

χ x ( u ) = 1 , u = x , 0 , u x.

Obviously, we have

( n rc ) k = 0 p n + rc , k ( x , c ) 0 x t f ( x ) f ( x + ) + f ( x ) 2 χ x ( u ) du p n ( r 1 ) c , k + r 1 ( t , c ) dt = 0 .

Thus, using the above identities, we can write

( Γ ( n c ) ) 2 Γ ( n c + r ) Γ ( n c r ) G n , r , c ( f ; x ) f ( x )

0 x t ( n rc ) k = 0 p n + rc , k ( x , c ) p n ( r 1 ) c , k + r 1 ( t , c ) × f ( x + ) + f ( x ) 2 + ( f ) x ( u ) du dt + 0 x t ( n rc ) k = 0 p n + rc , k ( x , c ) p n ( r 1 ) c , k + r 1 ( t , c ) [ f ( x + ) f ( x ) ] 2 sgn ( u x ) du dt . (3.1)

Also it can be verified that

0 x t [ f ( x + ) f ( x ) ] 2 sgn ( u x ) du ( n rc ) k = 0 p n + rc , k ( x , c ) p n ( r 1 ) c , k + r 1 ( t , c ) dt | f ( x + ) f ( x ) | 2 [ T n , r , 2 ( x , c ) ] 1 / 2 (3.2)

and

0 x t [ f ( x + ) + f ( x ) ] 2 du ( n rc ) k = 0 p n + rc , k ( x , c ) p n ( r 1 ) c , k + r 1 ( t , c ) dt | f ( x + ) + f ( x ) | 2 T n , r , 1 ( x , c ) . (3.3)

Combining Equations 3.1–3.3, we get

( Γ ( n c ) ) 2 Γ ( n c + r ) Γ ( n c r ) G n , r , c ( f ; x ) f ( x )

x x t ( f ) x ( u ) du ( n rc ) k = 0 p n + rc , k ( x , c ) p n ( r 1 ) c , k + r 1 ( t , c ) dt + 0 x x t ( f ) x ( u ) du ( n rc ) k = 0 p n + rc , k ( x , c ) p n ( r 1 ) c , k + r 1 ( t , c ) dt + | f ( x + ) f ( x ) | 2 [ T n , r , 2 ( x , c ) ] 1 / 2 + | f ( x + ) + f ( x ) | 2 T n , r , 1 ( x , c )

= | A n , r ( f , x ) + B n , r ( f , x ) | + | f ( x + ) f ( x ) | 2 [ T n , r , 2 ( x , c ) ] 1 / 2 + | f ( x + ) + f ( x ) | 2 T n , r , 1 ( x , c ) . (3.4)

Applying Remark 2 and Lemma 1 in Equation 3.4, we have

( Γ ( n c ) ) 2 Γ ( n c + r ) Γ ( n c r ) G n , r , c ( f ; x ) f ( x )

| A n , r ( f , x ) | + | B n , r ( f , x ) | + | f ( x + ) f ( x ) | 2 λx ( 1 + cx ) n + | f ( x + ) + f ( x ) | 2 r ( 1 + 2 cx ) + cx n ( r + 1 ) c . (3.5)

In order to complete the proof of the theorem, it suffices to estimate the terms An,r(f,x) and Bn,r(f,x) as follows:

| A n , r ( f , x ) | = x x t ( f ) x ( u ) du ( n rc ) k = 0 p n + rc , k ( x , c ) p n ( r 1 ) c , k + r 1 ( t , c ) dt = 2 x x t ( f ) x ( u ) du ( n rc ) k = 0 p n + rc , k ( x , c ) p n ( r 1 ) c , k + r 1 ( t , c ) dt + x 2 x x t ( f ) x ( u ) du ( n rc ) k = 0 p n + rc , k ( x , c ) p n ( r 1 ) c , k + r 1 ( t , c ) dt ( n rc ) k = 0 p n + rc , k ( x , c ) 2 x f ( t ) f ( x ) p n ( r 1 ) c , k + r 1 ( t , c ) dt + | f ( x + ) | ( n rc ) k = 0 p n + rc , k ( x , c ) x 2 x p n ( r 1 ) c , k + r 1 ( t , c ) ( t x ) dt + x 2 x ( f ) x ( u ) du 1 μ n , r ( x , 2 x ) + x 2 x | ( f ) x ( t ) | · 1 μ n , r ( x , t ) dt

( n rc ) k = 0 p n + rc , k ( x , c ) 2 x p n ( r 1 ) c , k + r 1 ( t , c ) C 1 t q dt + | f ( x ) | x 2 ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) ( t x ) 2 dt

+ | f ( x + ) | ( n rc ) k = 0 p n + rc , k ( x , c ) 2 x p n ( r 1 ) c , k + r 1 ( t , c ) | t x | dt + λ ( 1 + cx ) nx ( | f ( 2 x ) f ( x ) x f ( x + ) | + λ ( 1 + cx ) n k = 1 [ n ] x x k x + x k ( ( f ) x ) + x n x x n x + x n ( ( f ) x ) . (3.6)

For estimating the integral ( n rc ) k = 0 p n + rc , k ( x , c ) 2 x p n ( r 1 ) c , k + r 1 ( t , c ) C 1 t q dt above, we proceed as follows: since t≥2x implies that t≤2(tx) Schwarz inequality which follows from Lemma 1,

( n rc ) k = 0 p n + rc , k ( x , c ) 2 x p n ( r 1 ) c , k + r 1 ( t , c ) C 1 t q dt C 1 2 q ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) C 1 ( t x ) q dt = C 1 2 q T n , r , q ( x , c ) = O ( n q / 2 ) , as n and (3.7)

| f ( x ) | x 2 ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) × ( t x ) 2 dt = | f ( x ) | λ ( 1 + cx ) nx . (3.8)

By using the Hölder’s inequality and Remark 2, we get the estimate as follows:

| f ( x + ) | ( n rc ) k = 0 p n + rc , k ( x , c ) 2 x p n ( r 1 ) c , k + r 1 ( t , c ) | t x | dt

| f ( x + ) | ( n rc ) k = 0 p n + rc , k ( x , c ) 0 p n ( r 1 ) c , k + r 1 ( t , c ) ( t x ) 2 dt 1 / 2 = | f ( x + ) | λx ( 1 + cx ) n . (3.9)

Collecting the estimates from Equations 3.6–3.9, we obtain

| A n , r ( f , x ) | = O ( n q ) + | f ( x + ) | λx ( 1 + cx ) n + λ ( 1 + cx ) nx ( | f ( 2 x ) f ( x ) x f ( x + ) | + | f ( x ) | ) + λ ( 1 + cx ) n k = 1 [ n ] x x k x + x k ( ( f ) x ) + x n x x n x + x n ( ( f ) x ) . (3.10)

On the other hand, to estimate Bn,r(f,x) by applying the Lemma 2 with y = x x n and integration by parts, we have

| B n , r ( f , x ) | = 0 x x t ( f ) x ( u ) d t ( μ n , r ( x , t ) ) = 0 x ( f ) x ( u ) du μ n , r ( x , t ) 0 y + y x | ( f ) x ( t ) | | μ n , r ( x , t ) | dt

λx ( 1 + cx ) n 0 y t x ( ( f ) x ) 1 ( x t ) 2 dt + y x t x ( ( f ) x ) dt λx ( 1 + cx ) n 0 y t x ( ( f ) x ) 1 ( x t ) 2 dt + x n x x n x ( ( f ) x ) = λx ( 1 + cx ) n 1 n x x u x ( ( f ) x ) du + x n x x n x ( ( f ) x ) λ ( 1 + cx ) n k = 1 [ n ] x x k x ( ( f ) x ) + x n x x n x ( ( f ) x ) , (3.11)

where u = x x t .

Through combining the Equations 3.4, 3.10, and 3.11, we get the desired results. □

As a consequence of Lemma 3, we can easily prove the following corollary for the derivatives of the operators Gn,r,c.

Corollary 1

Let fsDBq(0,), q>0 and x∈(0,). The for λ>2 and n sufficiently large, we have

( Γ ( n c ) ) 2 Γ ( n c + r ) Γ ( n c r ) D s G n , r , c ( f ; x ) f s ( x )

λ ( 1 + cx ) n k = 1 [ n ] x x / k x + x / k ( ( D s + 1 f ) x ) + x n x x / n x + x / n ( ( D s + 1 f ) x ) + λ ( 1 + cx ) n ( | D s f ( 2 x ) D s f ( x ) x D s + 1 f ( x + ) | + | D s f ( x ) | ) + O ( n q ) + λ ( 1 + cx ) n | D s + 1 f ( x + ) | + | D s + 1 f ( x + ) D s + 1 f ( x ) | 2 λx ( 1 + cx ) n + | D s + 1 f ( x + ) + D s + 1 f ( x ) | 2 r ( 1 + 2 cx ) + cx n ( r + 1 ) c ,

where a b f x denotes the total variation of fx on [a,b], and the auxiliary function Ds + 1fx is defined by

D s + 1 f x ( t ) = D s + 1 f ( t ) D s + 1 f ( x ) , 0 t < x , 0 , t = x , D s + 1 f ( t ) D s + 1 f ( x + ) , x < t < ∞.

Conclusions

We have obtained the rate of convergence for the generalized Srivastava-Gupta operators for the functions having the derivatives of bounded variation which gives a better rate of convergence than the classical Srivastava-Gupta operators.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

DKV and PNA contributed equally to this work. Both authors read and approved the final manuscript.

Authors’ information

DKV is a research fellow at the Department of Mathematics, Indian Institute of Technology Roorkee in Roorkee, India. PNA is a professor at the Department of Mathematics, Indian Institute of Technology Roorkee in Roorkee, India.

Acknowledgements

The authors are thankful to the referees for valuable suggestions, leading to an overall improvement in the paper. The first author is also thankful to the Ministry of Human Resource and Development India for the financial support to carry out the above work.

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