Statistical approximation by Kantorovich-type discrete q-Betaoperators

Abstract

The aim of the present paper is to introduce a Kantorovich-type modification ofthe q-discrete beta operators and to investigate their statistical andweighted statistical approximation properties. Rates of statistical convergenceby means of the modulus of continuity and the Lipschitz-type function are alsoestablished for operators. Finally, we construct a bivariate generalization ofthe operator and also obtain the statistical approximation properties.

MSC: 41A25, 41A36.

1 Introduction

Gupta et al.[1] introduced discrete q-Beta operators as follows:

$$V_{n,q}(f(t);x)=V_n(f;q;x)=\frac{1}{{[n]}_q}\sum _{k=0}^{\mathrm{\infty }}{p}_{n,k}(q;x)f\left(\frac{{[k]}_q}{{[n+1]}_q{q}^{k-1}}\right),$$

(1.1)

where

$$p_{n,k}(q;x)=\frac{q^{k(k-1)/2}}{B_q(k+1,n)}\frac{x^k}{{(1+x)}_q^{n+k+1}}.$$

In the above paper, Gupta et al.[1] introduced and studied some approximation properties of these operators.They also obtained some global direct error estimates for the above operators usingthe second-order Ditzian-Totik modulus of smoothness and defined and studied thelimit discrete q-Beta operator. Also, they gave the following equalities:

$$\begin{array}{c}{V}_n(1;q;x)=1,V_n(t;q;x)=x\text{for every }n\in \mathbb{N}\text{and}\hfill \\ V_n(t^2;q;x)=(\frac{1}{q{[n+1]}_q}+1)x^2+\frac{x}{{[n+1]}_q}.\hfill \end{array}$$

In the recent years, applications of q-calculus in approximation theory isone of the interesting areas of research. Several authors have proposed theq analogues of Kantorovich-type modification of different linearpositive operators and studied their statistical approximation behaviors.

In 1974, Khan [2] studied approximation of functions in various classes using differenttypes of operators.

On the other hand, statistical convergence was first introduced by Fast [3], and it has become an area of active research. Also, statisticalconvergence was introduced by Gadjiev and Orhan [4], Doğru [5], Duman [6], Gupta and Radu [7], Ersan and Doğru [8] and Doğru and Örkcü [9].

In 2011, Örkcü and Doğru obtained weighted statistical approximationproperties of Kantorovich-type q-Szász-Mirakjan operators [10].

Recently, Doğru and Kanat [11] defined the Kantorovich-type modification of Lupas operators as follows:

$${\tilde{R}}_n(f;q;x)=[n+1]\sum _{k=0}^n({\int }_{[k]/[n+1]}^{[k+1]/[n+1]}f(t)d_{q}t)\left(\begin{array}{c}n\\ k\end{array}\right)\frac{q^{-k}{q}^{k(k-1)/2}{x}^k{(1-x)}^{n-k}}{(1-x+qx)\cdots (1-x+q^{n-1}x)}.$$

(1.2)

In [11], Doğru and Kanat proved the following statistical Korovkin-typeapproximation theorem for operators (1.2).

Theorem 1 Let$q:=(q_n)$, $0<q<1$, be a sequence satisfying the followingconditions:

$$\mathit{st}\text{-}\underset{n}{lim}{q}_n=1,\mathit{st}\text{-}\underset{n}{lim}{q}_n^n=a(a<1)\mathit{\text{and}}\mathit{st}\text{-}\underset{n}{lim}\frac{1}{[n]}=0,$$

(1.3)

then if f is any monotone increasing function defined on$[0,1]$, for the positive linear operator${\tilde{R}}_n(f;q;x)$, then

$$\mathit{st}\text{-}\underset{n}{lim}{\parallel {\tilde{R}}_n(f;q;\cdot )-f\parallel }_{C[0,1]}=0$$

holds.

In [5], Doğru gave some example so that $(q_n)$ is statistically convergent to 1 in ordinary case.Throughout the present paper, we consider $0<q<1$. Following [12, 13], for each non-negative integer n, we have

$$\begin{array}{c}{[n]}_q=\{\begin{array}{ll}\frac{1-q^n}{1-q},& q\ne 1,\\ n,& q=1,\end{array}\hfill \\ {[n]}_q!=\{\begin{array}{ll}{[n]}_q{[n-1]}_q{[n-2]}_q\cdots {[1]}_q,& n=1,2,\dots ,\\ 1,& n=0,\end{array}\hfill \end{array}$$

and

$${\left(\begin{array}{c}n\\ k\end{array}\right)}_q=\frac{{[n]}_q!}{{[k]}_q!{[n-k]}_q!}.$$

Further, we use the q-Pochhammer symbol, which is defined as

$${(1+x)}_q^n=\prod _{j=0}^{n-1}(1+q^{j}x).$$

The q-derivative ${\mathcal{D}}_{q}f$ of a function f is defined by

$$({\mathcal{D}}_{q}f)(x)=\frac{f(x)-f(qx)}{(1-q)x}\text{if }x\ne 0.$$

The q-Jackson integral is defined as (see [14])

$${\int }_0^{b}f(x)d_{q}x=(1-q)b\sum _{n=0}^{\mathrm{\infty }}f\left(b{q}^n\right)q^n,0<q<1,$$

(1.4)

and over a general interval $[a,b]$, one defines

$${\int }_a^{b}f(x)d_{q}x={\int }_0^{b}f(x)d_{q}x-{\int }_0^{a}f(x)d_{q}x.$$

(1.5)

Now, let us consider the following Kantorovich-type modification of discreteq-Beta operators for each positive integer n and$q\in (0,1)$:

$$V_n^{\ast }(f;q;x)=\frac{{[n+1]}_q}{{[n]}_q}\sum _{k=0}^{\mathrm{\infty }}({\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}f(t)d_{q}t)\frac{p_{n,k}(q;x)}{q^{2k-1}},$$

(1.6)

where f is a continuous and non-decreasing function on the interval$[0,\mathrm{\infty })$, $x\in [0,\mathrm{\infty })$.

It is seen that the operators $V_n^{\ast }$ are linear from the definition ofq-integral, and since f is a non-decreasing function,q-integral is positive, so $V_n^{\ast }$ are positive.

To obtain the statistical convergence of operators (1.6), we need the following basicresult.

2 Basic result

Lemma 1 The following equalities hold:

• (i) $V_n^{\ast }(1;q;x)=1$,

• (ii) $V_n^{\ast }(t;q;x)=x+\frac{q}{{[2]}_q{[n+1]}_q}$,

• (iii) $V_n^{\ast }(t^2;q;x)=\frac{q^{n-2}{[n+2]}_q{x}^2}{{[n+1]}_q}{x}^2+(\frac{q^{n-1}}{{[n+1]}_q}+\frac{(2q+1)}{{[n+1]}_q{[3]}_q})x+\frac{q}{{[n+1]}_q^2{[3]}_q}$.

Proof By using (1.4), (1.5) and the equality ${[k+1]}_q=1+{[k]}_q$, we have

$${\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}{d}_{q}t=\frac{q^k}{{[n+1]}_q},$$

(2.1)

$${\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}t{d}_{q}t=\frac{q^k}{{[n+1]}_q^2}({[k]}_q+\frac{1}{{[2]}_q}),$$

(2.2)

$${\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}{t}^2{d}_{q}t=\frac{q^k}{{[n+1]}_q^3}({[k]}_q^2+\frac{2q+1}{{[3]}_q}{[k]}_q+\frac{1}{{[3]}_q}).$$

(2.3)

Hence, by using $V_n(1;q;x)=1$ and (2.2), we get

$$V_n^{\ast }(1;q;x)=1.$$

Similarly, using (2.2), $V_n(1;q;x)=1$ and $V_n(t;q;x)=x$, we obtain

$$\begin{array}{rcl}{V}_n^{\ast }(t;q;x)& =& V_n(t;q;x)+V_n(1;q;x)\frac{q}{{[n+1]}_q^2{[3]}_q}\\ =& x+\frac{q}{{[2]}_q{[n+1]}_q}.\end{array}$$

Finally, using (2.3), $V_n(1;q;x)=1$, $V_n(t;q;x)=x$ and $V_n(t^2;q;x)=(\frac{1}{q{[n+1]}_q}+1)x^2+\frac{x}{{[n+1]}_q}$, we obtain

$$\begin{array}{rcl}{V}_n^{\ast }(t;q;x)& =& q^{n-1}{V}_n(t^2;q;x)+\frac{(2q+1)}{{[n+1]}_q{[3]}_q}{V}_n(t,q;x)+\frac{q}{{[n+1]}_q^2{[3]}_q}{V}_n(1,q;x)\\ =& \frac{q^{n-2}{[n+2]}_q}{{[n+1]}_q}{x}^2+(\frac{q^{n-1}}{{[n+1]}_q}+\frac{(2q+1)}{{[n+1]}_q{[3]}_q})x+\frac{q}{{[n+1]}_q^2{[3]}_q}.\end{array}$$

 □

Remark 1 From Lemma 1, we have

$$\begin{array}{c}{\alpha }_n(x)=V_n^{\ast }(t-x;q;x)=\frac{q}{{[2]}_q{[n+1]}_q},\hfill \\ \begin{array}{rl}{\delta }_n(x)=& V_n^{\ast }({(t-x)}^2;q;x)=V_n^{\ast }(t^2;q;x)-2x{V}_n^{\ast }(t;q;x)+x^2\\ =& (\frac{q^{n-2}{[n+2]}_q}{{[n+1]}_q}-1)x^2+(\frac{q^{n-1}}{{[n+1]}_q}+\frac{(2q+1)}{{[n+1]}_q{[3]}_q}-\frac{2q}{{[n+1]}_q{[2]}_q})x\\ +\frac{q}{{[n+1]}_q^2{[3]}_q}.\end{array}\hfill \end{array}$$

Remark 2 If we put $q=1$, we get the moments of Kantorovich-type modificationof discrete beta operators as

$$\begin{array}{c}{V}_n^{\ast }(t;1;x)=x+\frac{1}{2(n+1)},\hfill \\ V_n^{\ast }(t^2;1;x)=\frac{(n+2)}{(n+1)}{x}^2+\frac{2x}{(n+1)}+\frac{1}{3{(n+1)}^2},\hfill \\ V_n^{\ast }(t-x;1;x)=\frac{1}{2(n+1)},\hfill \end{array}$$

and

$$\begin{array}{rcl}{V}_n^{\ast }({(t-x)}^2;1;x)& =& V_n^{\ast }(t^2;1;x)-2x{V}_n^{\ast }(t;1;x)+x^2\\ =& \frac{x^2}{(n+1)}+\frac{x}{(n+1)}+\frac{1}{3{(n+1)}^2}.\end{array}$$

3 Korovkin-type statistical approximation properties

The study of Korovkin-type statistical approximation theory is a well-establishedarea of research, which deals with the problem of approximating a function with thehelp of a sequence of positive linear operators (see [9, 15, 16] for details). The usual Korovkin theorem is devoted to approximation bypositive linear operators on finite intervals. The main aim of this paper is toobtain the Korovkin-type statistical approximation properties of our operatorsdefined in (1.6), with the help of Theorem 1.

Let us recall the concept of a limit of a sequence extended to a statistical limit byusing the natural density δ of a set K of positive integers:

$$\delta (K)=\underset{n}{lim}\frac{1}{n}\{\text{the number }k\le n\text{ such that }k\in K\}$$

whenever the limit exists (see [17]). So, the sequence $x=(x_k)$ is said to be statistically convergent to a numberL, meaning that for every $\epsilon >0$,

$$\delta \{k:|x_k-L|\ge \epsilon \}=0$$

and it is denoted by $\mathit{st}\text{-}{lim}_k{x}_k=L$.

In this part, we will use the notation $\parallel f\parallel$ instead of ${\parallel f\parallel }_{C[0,\mu ]}$ for abbreviation.

Theorem 2 Let$q=(q_n)$be a sequence satisfying (1.3) for$0<q_n\le 1$and a Kantorovich-type modification of discrete q-Beta operators given by (1.6). Then, for anyfunction$f\in C[0,\mu ]\subset C[0,\mathrm{\infty })$and$x\in [0,\mu ]\subset [0,\mathrm{\infty })$, where$\mu >0$, we have

$$\mathit{st}\text{-}\underset{n}{lim}\parallel V_n^{\ast }(f;q_n;\cdot )-f\parallel =0,$$

where$C[0,\mu ]$denotes the space of all real bounded functions f which are continuous in$[0,\mu ]$.

Proof Using $V_n^{\ast }(1;q_n;x)=1$, it is clear that

$$\mathit{st}\text{-}\underset{n}{lim}\parallel V_n^{\ast }(1;q_n;x)-1\parallel =0.$$

Now, by Lemma 1(ii), we have

$$\parallel V_n^{\ast }(t;q_n;x)-x\parallel =\parallel x+\frac{q_n}{{[2]}_{q_n}{[n+1]}_{q_n}}-x\parallel \le \frac{q_n}{{[2]}_{q_n}{[n+1]}_{q_n}}.$$

(3.1)

For a given $\epsilon >0$, we define the following sets:

$$U=\{k:\parallel V_n^{\ast }(t;q_k;x)-x\parallel \ge \epsilon \}$$

and

$$U_1=\{k:\frac{q_k}{{[2]}_{q_k}{[k+1]}_{q_k}}\ge \epsilon \}.$$

From (3.1), one can see that $U\subset U_1$. So, we get

$$\delta \{k\le n:\parallel V_n^{\ast }(t,q_k;x)-x\parallel \ge \epsilon \}\le \delta \{k\le n:\frac{q_k}{{[2]}_{q_k}{[k+1]}_{q_k}}\ge \epsilon \}.$$

By using (1.3), it is clear that

$$\mathit{st}\text{-}\underset{n}{lim}\left(\frac{q_n}{{[2]}_{q_n}{[n+1]}_{q_n}}\right)=0.$$

So,

$$\delta \{k\le n:\frac{q_k}{{[2]}_{q_k}{[k+1]}_{q_k}}\ge \epsilon \}=0,$$

then

$$\mathit{st}\text{-}\underset{n}{lim}\parallel V_n^{\ast }(t;q_n;x)-x\parallel =0.$$

Finally, by Lemma 1(iii), we have

$$\begin{array}{r}\parallel V_n^{\ast }(t^2;q_n;x)-x^2\parallel \\ =\parallel \frac{q_n^{n-2}{[n+2]}_{q_n}}{{[n+1]}_{q_n}}{x}^2+(\frac{q_n^{n-1}}{{[n+1]}_{q_n}}+\frac{(2q_n+1)}{{[n+1]}_{q_n}{[3]}_{q_n}})x+\frac{q_n}{{[n+1]}_{q_n}^2{[3]}_{q_n}}-x^2\parallel \\ \le |\frac{q_n^{n-2}{[n+2]}_{q_n}}{{[n+1]}_{q_n}}-1|{\mu }^2+|\frac{q_n^{n-1}}{{[n+1]}_{q_n}}+\frac{(2q_n+1)}{{[n+1]}_{q_n}{[3]}_{q_n}}|\mu +\left|\frac{q_n}{{[n+1]}_{q_n}^2{[3]}_{q_n}}\right|\\ =A_2((\frac{1}{q_n}-1)+\frac{1}{{[n+1]}_{q_n}}(q_n^{n-2}+q_n^{n-1}+\frac{(2q_n+1)}{{[3]}_{q_n}})+\frac{q_n}{{[n+1]}_{q_n}^2{[3]}_{q_n}}),\end{array}$$

where $A_2=max\{{\mu }^2,\mu ,1\}={\mu }^2$.

If we choose ${\alpha }_n=(\frac{1}{q_n}-1)$, ${\beta }_n=\frac{1}{{[n+1]}_{q_n}}(q_n^{n-2}+q_n^{n-1}+\frac{(2q_n+1)}{{[3]}_{q_n}})$, ${\gamma }_n=\frac{q_n}{{[n+1]}_{q_n}^2{[3]}_{q_n}}$, then one can write

$$\mathit{st}\text{-}\underset{n}{lim}{\alpha }_n=\mathit{st}\text{-}\underset{n}{lim}{\beta }_n=\mathit{st}\text{-}\underset{n}{lim}{\gamma }_n=0,$$

(3.2)

by (1.3). Now, given $\epsilon >0$, we define the following four sets:

$$\begin{array}{c}S=\{k:\parallel V_n^{\ast }(t^2;q_k;x)-x^2\parallel \ge \frac{\epsilon }{A_2}\},\hfill \\ S_1=\{k:{\alpha }_k\ge \frac{\epsilon }{3A_2}\},S_2=\{k:{\beta }_k\ge \frac{\epsilon }{3A_2}\},S_3=\{k:{\gamma }_k\ge \frac{\epsilon }{3A_2}\}.\hfill \end{array}$$

It is obvious that $S\subseteq S_1\cup S_2\cup S_3$. So, we get

$$\begin{array}{rcl}\delta \{k\le n:\parallel V_n^{\ast }(t;q_k;x)-x\parallel \ge \frac{\epsilon }{A_2}\}& \le & \delta \{k\le n:{\alpha }_k\ge \frac{\epsilon }{3A_2}\}+\delta \{k\le n:{\beta }_k\ge \frac{\epsilon }{3A_2}\}\\ +\delta \{k\le n:{\gamma }_k\ge \frac{\epsilon }{3A_2}\}.\end{array}$$

So, the right-hand side of the inequalities is zero by (3.2), then

$$\mathit{st}\text{-}\underset{n}{lim}\parallel V_n^{\ast }(t^2;q_n;x)-x^2\parallel =0.$$

So, the proof is completed. □

4 Weighted statistical approximation

Let $B_{x^2}[0,\mathrm{\infty })$ be the set of all functions f defined on$[0,\mathrm{\infty })$ satisfying the condition $f(x)\le M_f(1+x^2)$, where $M_f$ is a constant depending only on f. By$C_{x^2}[0,\mathrm{\infty })$, we denote the subspace of all continuous functionsbelonging to $B_{x^2}[0,\mathrm{\infty })$. Also, let $C_{x^2}^{\ast }[0,\mathrm{\infty })$ be the subspace of all functions$f\in C_{x^2}[0,\mathrm{\infty })$, for which ${lim}_{x\to \mathrm{\infty }}\frac{f(x)}{1+x^2}$ is finite. The norm on $C_{x^2}^{\ast }[0,\mathrm{\infty })$ is ${\parallel f\parallel }_{x^2}={sup}_{x\in [0,\mathrm{\infty })}\frac{|f(x)|}{1+x^2}$.

Theorem 3 Let$q=(q_n)$be a sequence satisfying (1.3) for$0<q_n\le 1$. Then, for all nondecreasingfunctions$f\in C_{x^2}^{\ast }[0,\mathrm{\infty })$, we have

$$\mathit{st}\text{-}\underset{n}{lim}{\parallel V_n^{\ast }(f;q_n;\cdot )-f\parallel }_{x^2}=0.$$

Proof As a consequence of Lemma 1, since $V_n^{\ast }(x^2;q_n;x)\le C_1{x}^2$, where $C_1$ is a positive constant, $V_n^{\ast }(f;q_n;x)$ is a sequence of linear positive operators actingfrom $C_{x^2}^{\ast }[0,\mathrm{\infty })$ to $B_{x^2}[0,\mathrm{\infty })$. Using $V_n^{\ast }(1;q_n;x)=1$, it is clear that

$$\mathit{st}\text{-}\underset{n}{lim}{\parallel V_n^{\ast }(1;q_n;x)-1\parallel }_{x^2}=0.$$

Now, by Lemma 1(ii), we have

$${\parallel V_n^{\ast }(t;q_n;x)-x\parallel }_{x^2}=\underset{x\in [0,\mathrm{\infty })}{sup}\frac{|V_n^{\ast }(t;q_n;x)-x|}{1+x^2}\le \frac{q_n}{{[2]}_{q_n}{[n+1]}_{q_n}}.$$

(4.1)

By using (1.3), it is clear that

$$\mathit{st}\text{-}\underset{n}{lim}\left(\frac{q_n}{{[2]}_{q_n}{[n+1]}_{q_n}}\right)=0,$$

then

$$\mathit{st}\text{-}\underset{n}{lim}{\parallel V_n^{\ast }(t;q_n;x)-x\parallel }_{x^2}=0.$$

Finally, by Lemma 1(iii), we have

$$\begin{array}{r}{\parallel V_n^{\ast }(t^2;q_n;x)-x^2\parallel }_{x^2}\\ \le (\frac{q_n^{n-2}{[n+2]}_{q_n}}{{[n+1]}_{q_n}}-1)\underset{x\in [0,\mathrm{\infty })}{sup}\frac{x^2}{1+x^2}+(\frac{q_n^{n-1}}{{[n+1]}_{q_n}}+\frac{(2q_n+1)}{{[n+1]}_{q_n}{[3]}_{q_n}})\\ \times \underset{x\in [0,\mathrm{\infty })}{sup}\frac{x}{1+x^2}+\frac{q_n}{{[n+1]}_{q_n}^2{[3]}_{q_n}}\\ \le (\frac{q_n^{n-2}{[n+2]}_{q_n}}{{[n+1]}_{q_n}}-1)+(\frac{q_n^{n-1}}{{[n+1]}_{q_n}}+\frac{(2q_n+1)}{{[n+1]}_{q_n}{[3]}_{q_n}})+\frac{q_n}{{[n+1]}_{q_n}^2{[3]}_{q_n}}\\ =(\frac{1}{q_n}-1)+\frac{1}{{[n+1]}_{q_n}}(q_n^{n-2}+q_n^{n-1}+\frac{(2q_n+1)}{{[3]}_{q_n}})+\frac{q_n}{{[n+1]}_{q_n}^2{[3]}_{q_n}}.\end{array}$$

If we choose ${\alpha }_n=(\frac{1}{q_n}-1)$, ${\beta }_n=\frac{1}{{[n+1]}_{q_n}}(q_n^{n-2}+q_n^{n-1}+\frac{(2q_n+1)}{{[3]}_{q_n}})$, ${\gamma }_n=\frac{q_n}{{[n+1]}_{q_n}^2{[3]}_{q_n}}$, then one can write

$$\mathit{st}\text{-}\underset{n}{lim}{\alpha }_n=\mathit{st}\text{-}\underset{n}{lim}{\beta }_n=\mathit{st}\text{-}\underset{n}{lim}{\gamma }_n=0,$$

(4.2)

by (1.3). Now, given $\epsilon >0$, we define the following four sets:

$$\begin{array}{c}P=\{k:{\parallel V_n^{\ast }(t^2;q_k;x)-x^2\parallel }_{x^2}\ge \epsilon \},\hfill \\ P_1=\{k:{\alpha }_k\ge \frac{\epsilon }{3}\},P_2=\{k:{\beta }_k\ge \frac{\epsilon }{3}\},P_3=\{k:{\gamma }_k\ge \frac{\epsilon }{3}\}.\hfill \end{array}$$

It is obvious that $P\subseteq P_1\cup P_2\cup P_3$. So, we get

$$\begin{array}{rcl}\delta \{k\le n:{\parallel V_n^{\ast }(t;q_k;x)-x\parallel }_{x^2}\ge \epsilon \}& \le & \delta \{k\le n:{\alpha }_k\ge \frac{\epsilon }{3}\}+\delta \{k\le n:{\beta }_k\ge \frac{\epsilon }{3}\}\\ +\delta \{k\le n:{\gamma }_k\ge \frac{\epsilon }{3}\}.\end{array}$$

So, the right-hand side of the inequalities is zero by (4.2), then

$$\mathit{st}\text{-}\underset{n}{lim}{\parallel V_n^{\ast }(t^2;q_n;x)-x^2\parallel }_{x^2}=0.$$

So, the proof is completed. □

5 Rates of statistical convergence

In this part, rates of statistical convergence of operator (1.6) by means of modulusof continuity and Lipschitz functions are introduced.

Lemma 2[15]

Let$0<q<1$and$a\in [0,bq]$, $b>0$. The inequality

$${\int }_a^b|t-x|d_{q}t\le {\left({\int }_a^b{|t-x|}^2{d}_{q}t\right)}^{1/2}{\left({\int }_a^b{d}_{q}t\right)}^{1/2}$$

is satisfied.

Let $C_B[0,\mathrm{\infty })$, the space of all bounded and continuous functions on$[0,\mathrm{\infty })$, and $x\ge 0$. Then, for $\delta >0$, the modulus of continuity of f denoted by$\omega (f;\delta )$ is defined to be

$$\omega (f;\delta )=\underset{x-\delta \le t\le x+\delta ;t\in [0,\mathrm{\infty })}{sup}|f(t)-f(x)|.$$

Then it is known that ${lim}_{\delta \to 0}\omega (f;\delta )=0$ for $f\in C_B[0,\mathrm{\infty })$, and also, for any $\delta >0$ and each $t,x\ge 0$, we have

$$|f(t)-f(x)|\le \omega (f;\delta )(1+\frac{|t-x|}{\delta }).$$

(5.1)

Theorem 4 Let$(q_n)$be a sequence satisfying (1.3). For every non-decreasing$f\in C_B[0,\mathrm{\infty })$, $x\ge 0$and$n\in \mathbb{N}$, we have

$$|V_n^{\ast }(f;q_n;x)-f(x)|\le 2\omega (f;\sqrt{{\delta }_n(x)}),$$

where

$$\begin{array}{rl}{\delta }_n(x)=& (\frac{q_n^{n-2}{[n+2]}_{q_n}}{{[n+1]}_{q_n}}-1)x^2+(\frac{q_n^{n-1}}{{[n+1]}_{q_n}}+\frac{(2q_n+1)}{{[n+1]}_{q_n}{[3]}_{q_n}}-\frac{2q_n}{{[n+1]}_{q_n}{[2]}_{q_n}})x\\ +\frac{q_n}{{[n+1]}_{q_n}^2{[3]}_{q_n}}.\end{array}$$

(5.2)

Proof Let non-decreasing $f\in C_B[0,\mathrm{\infty })$ and $x\ge 0$. Using linearity and positivity of the operators$V_n^{\ast }(f;q_n;x)$ and then applying (5.1), we get for$\delta >0$ and $n\in \mathbb{N}$ that

$$\begin{array}{rcl}|V_n^{\ast }(f;q_n;x)-f(x)|& \le & V_n^{\ast }(|f(t)-f(x)|;q_n;x)\\ \le & \omega (f;\delta )\{V_n^{\ast }(1;q_n;x)+\frac{1}{\delta }{V}_n^{\ast }(|t-x|;q_n;x)\}.\end{array}$$

Taking into account $V_n^{\ast }(1;q_n;x)=1$ and then applying Lemma 2 with$a={[k]}_q/{[n+1]}_q$ and $b={[k+1]}_q/{[n+1]}_q$, we may write

$$\begin{array}{rcl}|V_n^{\ast }(f;q_n;x)-f(x)|& \le & \omega (f;\delta )\{1+\frac{1}{\delta }\frac{{[n+1]}_q}{{[n]}_q}\sum _{k=0}^{\mathrm{\infty }}\frac{p_{n,k}(q;x)}{q^{2k-1}}{\left({\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}{(t-x)}^2{d}_{q}t\right)}^{1/2}\\ \times {\left({\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}{d}_{q}t\right)}^{1/2}\}.\end{array}$$

By using the Cauchy-Schwarz inequality, we have

$$\begin{array}{rcl}|V_n^{\ast }(f;q_n;x)-f(x)|& =& \omega (f;\delta )\{1+\frac{1}{\delta }{\left(\frac{{[n+1]}_q}{{[n]}_q}\sum _{k=0}^{\mathrm{\infty }}\frac{p_{n,k}(q;x)}{q^{2k-1}}{\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}{(t-x)}^2{d}_{q}t\right)}^{1/2}\\ \times {\left(\frac{{[n+1]}_q}{{[n]}_q}\sum _{k=0}^{\mathrm{\infty }}\frac{p_{n,k}(q;x)}{q^{2k-1}}{\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}{d}_{q}t\right)}^{1/2}\}\\ \le & \omega (f;\delta )\{1+\frac{1}{\delta }{\left(V_n^{\ast }({(t-x)}^2;q_n;x)\right)}^{1/2}{(V_n^{\ast }(1;q_n;x))}^{1/2}\}.\end{array}$$

Taking $q=q_n$, a sequence satisfying (1.3), and using${\delta }_n(x)=V_n^{\ast }({(t-x)}^2;q_n;x)$ and then choosing $\delta ={\delta }_n(x)$ as in (5.2), the theorem is proved. □

Notice that by the conditions in (1.3), $\mathit{st}\text{-}{lim}_n{\delta }_n=0$. By (5.1), we have

$$\mathit{st}\text{-}\underset{n}{lim}\omega (f;{\delta }_n)=0.$$

This gives us the pointwise rate of statistical convergence of the operators$V_n^{\ast }(f;q_n;x)$ to $f(x)$.

Now we will study the rate of convergence of the operator $V_n^{\ast }(f;q_n;x)$ with the help of functions of the Lipschitz class${Lip}_M(\alpha )$, where $M>0$ and $0<\alpha \le 1$. Recall that a function $f\in C_B[0,\mathrm{\infty })$ belongs to ${Lip}_M(\alpha )$ if the inequality

$$|f(t)-f(x)|\le M{|t-x|}^{\alpha };\mathrm{\forall }t,x\in [0,\mathrm{\infty }).$$

We have the following theorem.

Theorem 5 Let the sequence$q=(q_n)$satisfy the conditions given in (1.3), and let$f\in {Lip}_M(\alpha )$, $x\ge 0$with$0<\alpha \le 1$. Then

$$|V_n^{\ast }(f;q_n;x)-f(x)|\le M{\delta }_n^{\alpha /2}(x),$$

(5.3)

where${\delta }_n(x)$is given as in (5.2).

Proof Since $V_n^{\ast }(f;q_n;x)$ are linear positive operators and$f\in {Lip}_M(\alpha )$, on $x\ge 0$ with $0<\alpha \le 1$, we can write

$$\begin{array}{rcl}|V_n^{\ast }(f;q_n;x)-f(x)|& \le & V_n^{\ast }(|f(t)-f(x)|;q_n;x)\\ \le & M{V}_n^{\ast }({|t-x|}^{\alpha };q_n;x).\end{array}$$

If we take $p=\frac{2}{\alpha }$, $q=\frac{2}{2-\alpha }$, applying Lemma 2 and Hölder’s inequality,we obtain

$$\begin{array}{c}\begin{array}{rl}|V_n^{\ast }(f;q_n;x)-f(x)|\le & M\frac{{[n+1]}_q}{{[n]}_q}\sum _{k=0}^{\mathrm{\infty }}\frac{p_{n,k}(q;x)}{q^{2k-1}}{\left({\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}{(t-x)}^2{d}_{q}t\right)}^{\alpha /2}\\ \times {\left({\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}{d}_{q}t\right)}^{(2-\alpha )/2}\},\end{array}\hfill \\ \begin{array}{rl}|V_n^{\ast }(f;q_n;x)-f(x)|=& M{\left(\frac{{[n+1]}_q}{{[n]}_q}\sum _{k=0}^{\mathrm{\infty }}\frac{p_{n,k}(q;x)}{q^{2k-1}}{\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}{(t-x)}^2{d}_{q}t\right)}^{\alpha /2}\\ \times {\left(\frac{{[n+1]}_q}{{[n]}_q}\sum _{k=0}^{\mathrm{\infty }}\frac{p_{n,k}(q;x)}{q^{2k-1}}{\int }_{{[k]}_q/{[n+1]}_q}^{{[k+1]}_q/{[n+1]}_q}{d}_{q}t\right)}^{(2-\alpha )/2}\\ \le & M{\left(V_n^{\ast }({(t-x)}^2;q_n;x)\right)}^{\alpha /2}{(V_n^{\ast }(1;q_n;x))}^{(2-\alpha )/2}\\ \le & M{\left(V_n^{\ast }({(t-x)}^2;q_n;x)\right)}^{\alpha /2}.\end{array}\hfill \end{array}$$

Taking ${\delta }_n(x)=(V_n^{\ast }({(t-x)}^2;q_n;x))$, as in (5.2), we get the desiredresult. □

6 The construct of the bivariate operators of Kantorovich type

The purpose of this part is to give a representation for the bivariate operators ofKantorovich type (1.6), introduce the statistical convergence of the operators tothe function f and show the rate of statistical convergence of theseoperators.

For $f:C([0,\mathrm{\infty })\times [0,\mathrm{\infty }))\to C([0,\mathrm{\infty })\times [0,\mathrm{\infty }))$ and $0<q_{n_1},q_{n_2}\le 1$, let us define the bivariate case of operators (1.6)as follows:

$$\begin{array}{r}{V}_{n_1,n_2}^{\ast }(f;q_{n_1},q_{n_2},x,y)\\ =\frac{{[n_1+1]}_{q_{n_1}}{[n_2+1]}_{q_{n_2}}}{{[n_1]}_{q_{n_1}}{[n_2]}_{q_{n_2}}}\\ \times \sum _{k_1=0}^{\mathrm{\infty }}\sum _{k_2=0}^{\mathrm{\infty }}({\int }_{{[k_1]}_{q_{n_1}}/{[n_1+1]}_{q_{n_1}}}^{{[k_1+1]}_{q_{n_1}}/{[n_1+1]}_{q_{n_1}}}{\int }_{{[k_2]}_{q_{n_2}}/{[n_2+1]}_{q_{n_2}}}^{{[k_2+1]}_{q_{n_2}}/{[n_2+1]}_{q_{n_2}}}f(s,t)d_{q_{n_1}}s{d}_{q_{n_2}}t)\\ \times \frac{p_{n_1,n_2}(q_{n_1},q_{n_2},x,y)}{q_{n_1}^{2k_1-1}{q}_{n_2}^{2k_2-1}},\end{array}$$

(6.1)

where

$$p_{n_1,n_2,k_1,k_2}(q_{n_1},q_{n_2},x,y)=\frac{q_{n_1}^{k_1(k_1-1)/2}{q}_{n_2}^{k_2(k_2-1)/2}}{B_{q_{n_1}}(k_1+1,n_1)B_{q_{n_2}}(k_2+1,n_2)}\frac{x^{k_1}{y}^{k_2}}{{(1+x)}_{q_{n_1}}^{n_1+k_1+1}{(1+y)}_{q_{n_2}}^{n_2+k_2+1}}.$$

In [18], Erkuş and Duman proved the statistical Korovkin-type approximationtheorem for the bivariate linear positive operators to the functions in space$H_{{\omega }_2}$.

In 2006, Doğru and Gupta [19] introduced a bivariate generalization of the q-MKZ operators andinvestigated its Korovkin-type approximation properties.

Recently, Ersan and Doğru [8] obtained the statistical Korovkin-type theorem and lemma for thebivariate linear positive operators defined in the space $H_{{\omega }_2}$ as follows.

Theorem 6[8]

Let$L_{n_1,n_2}$be the sequence of linear positive operators acting from$H_{{\omega }_2}(R_{+}^2)$into$C_B(R_{+})$, where$R_{+}=[0,\mathrm{\infty })$. Then, for any$f\in H_{{\omega }_2}$,

$$\mathit{st}\text{-}\underset{n_1,n_2}{lim}\parallel L_{n_1,n_2}(f)-f\parallel =0.$$

Lemma 3[8]

The bivariate operators defined in[8]satisfy the following items:

• (i) $L_{n_1,n_2}(f_0;q_{n_1},q_{n_2},x,y)=q_{n_1}{q}_{n_2}$,

• (ii) $L_{n_1,n_2}(f_1;q_{n_1},q_{n_2},x,y)=q_{n_1}{q}_{n_2}\frac{{[n_1]}_{q_{n_1}}}{{[n_1+1]}_{q_{n_1}}}\frac{x}{1+x}$,

• (iii) $L_{n_1,n_2}(f_2;q_{n_1},q_{n_2},x,y)=q_{n_1}{q}_{n_2}\frac{{[n_2]}_{q_{n_2}}}{{[n_2+1]}_{q_{n_2}}}\frac{y}{1+y}$,

• (iv) $L_{n_1,n_2}(f_3;q_{n_1},q_{n_2},x,y)$ = $q_{n_1}^3{q}_{n_2}\frac{{[n_1]}_{q_{n_1}}{[n_1-1]}_{q_{n_1}}}{{[n_1+1]}_{q_{n_1}}^2}\frac{x^2}{(1+x)(1+q_{n_1}x)}$ + $q_{n_1}{q}_{n_2}\frac{{[n_1]}_{q_{n_1}}}{{[n_1+1]}_{q_{n_1}}^2}\frac{x}{1+x}$ + $q_{n_1}{q}_{n_2}^3\frac{{[n_2]}_{q_{n_2}}{[n_2-1]}_{q_{n_2}}}{{[n_2+1]}_{q_{n_2}}^2}\frac{y^2}{(1+y)(1+q_{n_2}y)}$ + $q_{n_1}{q}_{n_2}\frac{{[n_2]}_{q_{n_2}}}{{[n_2+1]}_{q_{n_2}}^2}\frac{y}{1+y}$.

In order to obtain the statistical convergence of operator (6.1), we need thefollowing lemma.

Lemma 4 The bivariate operators defined in (6.1) satisfy the followingequalities:

• (i) $V_{n_1,n_2}^{\ast }(f_0;q_{n_1},q_{n_2},x,y)=1$,

• (ii) $V_{n_1,n_2}^{\ast }(f_1;q_{n_1},q_{n_2},x,y)=x+\frac{q_{n_1}}{{[2]}_{q_{n_1}}{[n_1+1]}_{q_{n_1}}}$,

• (iii) $V_{n_1,n_2}^{\ast }(f_2;q_{n_1},q_{n_2},x,y)=y+\frac{q_{n_2}}{{[2]}_{q_{n_2}}{[n_2+1]}_{q_{n_2}}}$,

• (iv) $V_{n_1,n_2}^{\ast }(f_3;q_{n_1},q_{n_2},x,y)$ = $\frac{q_{n_1}^{n_1-2}{[n_1+2]}_{q_{n_1}}}{{[n_1+1]}_{q_{n_1}}}{x}^2$ + $(\frac{q_{n_1}^{n_1-1}}{{[n_1+1]}_{q_{n_1}}}+\frac{(2q_{n_1}+1)}{{[n_1+1]}_{q_{n_1}}{[3]}_{q_{n_1}}})x$ + $\frac{q_{n_1}}{{[n_1+1]}_{q_{n_1}}^2{[3]}_{q_{n_1}}}$ + $\frac{q_{n_2}^{n_2-2}{[n_2+1]}_{q_{n_2}}}{{[n_2+1]}_{q_{n_2}}}{y}^2$ + $(\frac{q_{n_2}^{n_2-1}}{{[n_2+1]}_{q_{n_2}}}+\frac{(2q_{n_2}+1)}{{[n_2+1]}_{q_{n_2}}{[3]}_{q_{n_2}}})y$ + $\frac{q_{n_2}}{{[n_2+1]}_{q_{n_2}}^2{[3]}_{q_{n_2}}}$.

Proof By the help of the proofs for the bivariate operator in [20], the conditions may be easily obtained. So, the proof can be omitted.

Let $q=(q_{n_1})$ and $q=(q_{n_2})$ be the sequence that converges statistically to 1 butdoes not converge in ordinary sense, so for $0<q_{n_1},q_{n_2}\le 1$, it can be written as

$$\mathit{st}\text{-}\underset{n_1}{lim}{q}_{n_1}=\mathit{st}\text{-}\underset{n_2}{lim}{q}_{n_2}=1.$$

(6.2)

Now, under the condition in (6.2), let us show the statistical convergence ofbivariate operator (6.1) with the help of the proof of Theorem 2. □

Theorem 7 Let$q=(q_{n_1})$and$q=(q_{n_2})$be a sequence satisfying (6.2) for$0<q_{n_1},q_{n_2}\le 1$, and let$V_{n_1,n_2}^{\ast }$be a sequence of linear positive operators from$C(K)$into$C(K)$given by (1.6). Then, for any function$f\in C(K_1\times K_1)\subset C(K\times K)$and$x\in K_1\times K_1\subset K\times K$, where$K=[0,\mathrm{\infty })\times [0,\mathrm{\infty })$, $K_1=[0,\mu ]\times [0,\mu ]$, we have

$$\mathit{st}\text{-}\underset{n_1,n_2}{lim}{\parallel V_{n_1,n_2}^{\ast }(f)-f\parallel }_{C(K_1\times K_1)}=0.$$

Proof Using Lemma 4, the proof can be obtained similar to the proof ofTheorem 2. So, we shall omit this proof. □

7 Rates of convergence of bivariate operators

Let $K=[0,\mathrm{\infty })\times [0,\mathrm{\infty })$. Then the sup norm on $C_B(K)$ is given by

$$\parallel f\parallel =\underset{(x,y)\in K}{sup}|f(x,y)|,f\in C_B(K).$$

We consider the modulus of continuity ${\omega }_2(f;{\delta }_1,{\delta }_2)$ for bivariate case given by ${\delta }_1,{\delta }_2>0$,

$${\omega }_2(f;{\delta }_1,{\delta }_2)=\{sup|f(x^{\prime },y^{\prime })-f(x,y)|:(x^{\prime },y^{\prime }),(x,y)\in K\text{ and }|x^{\prime }-x|\le {\delta }_1,|y^{\prime }-y|\le {\delta }_2\}.$$

It is clear that a necessary and sufficient condition for a function$f\in C_B(K)$ is

$$\underset{{\delta }_1,{\delta }_2\to 0}{lim}\omega (f;{\delta }_1,{\delta }_2)=0,$$

and $\omega (f;{\delta }_1,{\delta }_2)$ satisfy the following condition:

$$|f(x^{\prime },y^{\prime })-f(x,y)|\le \omega (f;{\delta }_1,{\delta }_2)(1+\frac{|x^{\prime }-x|}{{\delta }_1})(1+\frac{|y^{\prime }-y|}{{\delta }_2})$$

(7.1)

for each $f\in C_B(K)$. Then observe that any function in$C_B(K)$ is continuous and bounded on K. Details ofthe modulus of continuity for bivariate case can be found in [21].

Now, the rate of statistical convergence of bivariate operator (6.1) by means ofmodulus of continuity in $f\in C_B(K)$ will be given in the following theorem.

Theorem 8 Let$q=(q_{n_1})$and$q=(q_{n_2})$be a sequence satisfying the condition in (6.2). So, wehave

$$|V_{n_1,n_2}^{\ast }(f;q_{n_1},q_{n_2},x,y)-f(x,y)|\le 4\omega (f;\sqrt{{\delta }_{n_1(x)}},\sqrt{{\delta }_{n_2(x)}}),$$

where

$$\begin{array}{rl}{\delta }_{n_1}(x)=& (\frac{q_{n_1}^{n_1-2}{[n_1+2]}_{q_{n_1}}}{{[n_1+1]}_{q_{n_1}}}-1)x^2+(\frac{q_{n_1}^{n_1-1}}{{[n_1+1]}_{q_{n_1}}}+\frac{(2q_{n_1}+1)}{{[n_1+1]}_{q_{n_1}}{[3]}_{q_{n_1}}}\\ -\frac{2q_{n_1}}{{[n_1+1]}_{q_{n_1}}{[2]}_{q_{n_1}}})x+\frac{q_{n_1}}{{[n_1+1]}_{q_{n_1}}^2{[3]}_{q_{n_1}}},\end{array}$$

(7.2)

$$\begin{array}{rl}{\delta }_{n_2}(y)=& (\frac{q_{n_2}^{n_2-2}{[n_2+2]}_{q_{n_2}}}{{[n_2+1]}_{q_{n_2}}}-1)y^2+(\frac{q_{n_2}^{n_2-1}}{{[n_2+1]}_{q_{n_2}}}+\frac{(2q_{n_2}+1)}{{[n_2+1]}_{q_{n_2}}{[3]}_{q_{n_2}}}\\ -\frac{2q_{n_2}}{{[n_2+1]}_{q_{n_2}}{[2]}_{q_{n_2}}})y+\frac{q_{n_2}}{{[n_2+1]}_{q_{n_2}}^2{[3]}_{q_{n_2}}}.\end{array}$$

(7.3)

Proof By using the condition in (7.1), we get for ${\delta }_{n_1},{\delta }_{n_2}>0$ and $n\in \mathbb{N}$ that

$$\begin{array}{r}|V_{n_1,n_2}^{\ast }(f;q_{n_1},q_{n_2},x,y)-f(x,y)|\\ \le V_{n_1,n_2}^{\ast }(|f(x^{\prime },y^{\prime })-f(x,y)|;q_{n_1},q_{n_2},x,y)\\ \le \omega (f;{\delta }_{n_1(x)},{\delta }_{n_2(x)})\{V_{n_1,n_2}^{\ast }(f_0;q_{n_1},q_{n_2},x,y)+\frac{1}{{\delta }_{n_1}}{V}_n^{\ast }(|x^{\prime }-x|;q_{n_1},q_{n_2},x,y)\}\\ \times \{V_{n_1,n_2}^{\ast }(f_0;q_{n_1},q_{n_2},x,y)+\frac{1}{{\delta }_{n_2}}{V}_n^{\ast }(|y^{\prime }-y|;q_{n_1},q_{n_2},x,y)\}.\end{array}$$

If the Cauchy-Schwarz inequality is applied, we have

$$V_n^{\ast }(|x^{\prime }-x|;q_{n_1},q_{n_2},x,y)\le {\left(V_n^{\ast }({(x^{\prime }-x)}^2;q_{n_1},q_{n_2},x,y)\right)}^{1/2}{(V_{n_1,n_2}^{\ast }(f_0;q_{n_1},q_{n_2},x,y))}^{1/2}.$$

So, if it is substituted in the above equation, the proof iscompleted. □

At last, the following theorem represents the rate of statistical convergence ofbivariate operator (6.1) by means of Lipschitz ${Lip}_M({\alpha }_1,{\alpha }_2)$ functions for the bivariate case, where$f\in C_B[0,\mathrm{\infty })$ and $M>0$ and $0<{\alpha }_1\le 1$, $0<{\alpha }_2\le 1$, then let us define ${Lip}_M({\alpha }_1,{\alpha }_2)$ as

$$|f(x^{\prime },y^{\prime })-f(x,y)|\le M{|x^{\prime }-x|}^{{\alpha }_1}{|y^{\prime }-y|}^{{\alpha }_2};\mathrm{\forall }x,x^{\prime },y,y^{\prime }\in [0,\mathrm{\infty }).$$

We have the following theorem.

Theorem 9 Let the sequence$q=(q_{n_1})$and$q=(q_{n_2})$satisfy the conditions given in (6.2), and let$f\in {Lip}_M({\alpha }_1,{\alpha }_2)$, $x\ge 0$and$0<{\alpha }_1\le 1$, $0<{\alpha }_2\le 1$. Then

$$|V_{n_1,n_2}^{\ast }(f;q_{n_1},q_{n_2},x,y)-f(x,y)|\le M{{\delta }_{n_1}}^{{\alpha }_1/2}(x){{\delta }_{n_2}}^{{\alpha }_2/2}(x),$$

where${\delta }_{n_1}(x)$and${\delta }_{n_2}(x)$are defined in (7.2), (7.3).

Proof Since $V_{n_1,n_2}^{\ast }(f;q_{n_1},q_{n_2},x,y)$ are linear positive operators and$f\in {Lip}_M({\alpha }_1,{\alpha }_2)$, $x\ge 0$ and $0<{\alpha }_1\le 1$, $0<{\alpha }_2\le 1$, we can write

$$\begin{array}{rcl}|V_{n_1,n_2}^{\ast }(f;q_{n_1},q_{n_2},x,y)-f(x,y)|& \le & V_{n_1,n_2}^{\ast }(|f(x^{\prime },y^{\prime })-f(x,y)|;q_{n_1},q_{n_2},x,y)\\ \le & M{V}_{n_1,n_2}^{\ast }({|x^{\prime }-x|}^{{\alpha }_1}{|y^{\prime }-y|}^{{\alpha }_2};q_{n_1},q_{n_2},x,y)\\ =& M{V}_{n_1,n_2}^{\ast }({|x^{\prime }-x|}^{{\alpha }_1};q_{n_1},q_{n_2},x,y)\\ \times V_{n_1,n_2}^{\ast }({|y^{\prime }-y|}^{{\alpha }_2};q_{n_1},q_{n_2},x,y).\end{array}$$

If we take $p_1=\frac{2}{{\alpha }_1}$, $p_2=\frac{2}{{\alpha }_2}$, $q_1=\frac{2}{2-{\alpha }_1}$, $q_2=\frac{2}{2-{\alpha }_2}$, applying Hölder’s inequality, we obtain

$$\begin{array}{rl}|V_{n_1,n_2}^{\ast }(f;q_{n_1},q_{n_2},x,y)-f(x,y)|\le & M{(V_{n_1,n_2}^{\ast }{(x^{\prime }-x)}^2;q_{n_1},q_{n_2},x,y)}^{{\alpha }_1/2}\\ \times {(V_{n_1,n_2}^{\ast }(f_0;q_{n_1},q_{n_2},x,y))}^{2-{\alpha }_1/2}\\ \times (V_{n_1,n_2}^{\ast }{({(y^{\prime }-y)}^{{\alpha }_2};q_{n_1},q_{n_2},x,y)}^{{\alpha }_2/2}\\ \times {(V_{n_1,n_2}^{\ast }(f_0;q_{n_1},q_{n_2},x,y))}^{2-{\alpha }_2/2}\\ =& M{{\delta }_{n_1}}^{{\alpha }_1/2}(x){{\delta }_{n_2}}^{{\alpha }_2/2}(x).\end{array}$$

So, the proof is completed. □

Authors’ information

Dr. VNM is an assistant professor at the Sardar Vallabhbhai National Institute ofTechnology, Ichchhanath Mahadev Road, Surat, Gujarat, India and he is a very activeresearcher in various fields of mathematics like Approximation theory, summabilitytheory, variational inequalities, fixed point theory and applications, operatoranalysis, nonlinear analysis etc. A Ph.D. in Mathematics, he is adouble gold medalist, ranking first in the order of merit in both B.Sc. and M.Sc.Examinations from the Dr. Ram Manohar Lohia Avadh University, Faizabad (UttarPradesh), India. Dr. VNM has undergone rigorous training from IIT, Roorkee, Mumbai,Kanpur; ISI Banglore in computer oriented mathematical methods and has experience ofteaching post graduate, graduate and engineering students. Dr. VNM has to his creditmany research publications in reputed journals including SCI/SCI(Exp.) accreditedjournals. Dr. VNM is referee of several international journals in the frame of pureand applied mathematics and Editor of reputed journals covering the subjectmathematics. The second author KK is a research scholar (R/S) in Applied Mathematicsand Humanities Department at the Sardar Vallabhbhai National Institute ofTechnology, Ichchhanath Mahadev Road, Surat (Gujarat), India under the guidance ofDr. VNM. Recently the third author LNM joined as a full-time research scholar at theDepartment of Mathematics, National Institute of Technology, Silchar-788010,District-Cachar, Assam, India and he is also very good active researcher inapproximation theory, summability analysis, integral equations, nonlinear analysis,optimization technique, fixed point theory and operator theory.