Skip to main content

\((p,q)\)-gamma operators which preserve \(x^{2}\)

Abstract

In this paper, we introduce \((p,q)\)-gamma operators which preserve \(x^{2}\), we estimate the moments of these operators, and establish direct and local approximation theorems of these operators. Then two approximation theorems about Lipschitz functions are obtained. The estimates on the rate of convergence and some weighted approximation theorems of the operators are also obtained. Furthermore, the Voronovskaja-type asymptotic formula is also presented.

Introduction

With the rapid development of the approximation theory about the operators since the last century, lots of operators, such as Bernstein operators [4], Szász–Mirakjan operators [32, 37], Baskakov operators [3], Bleimann–Butzer–Hann operators [5], and Meyer–König–Zeller operators [31], have been proposed and constructed by several researchers due to Weierstrass and the important convergence theorem of Korovkin [26], see also [17]. In [23], Karsli considered gamma operators and studied the rate of convergence of these operators for the functions with derivative of bounded variation

$$ L_{n}(f;x)=\frac{(2n+3)!x^{n+3}}{n!(n+2)!} \int _{0}^{\infty }\frac{t ^{n}}{(x+t)^{2n+4}}f(t)\,\mathrm{d}t,\quad x>0. $$
(1)

In [25], Karsli and Ozarslan established some local and global approximation results for the operators \(L_{n}\).

In recent years, with the rapid development of q-calculus [22], the study of new polynomials and operators constructed with q-integer has attracted more and more attention. Lupas first introduced q-Bernstein polynomials [27], and Phillips [36] proposed other q-analogue of Bernstein polynomials. Later, many researchers have performed studies in this field, and the q-analogue of classical operators and modified operators, such as q-Szász–Mirakjan operators [28], q-Baskakov operators [13], q-Meyer–König–Zeller operators [12], q-Bleimann–Butzer–Hann operators [11] q-Phillips operators [29], q-Baskakov–Kantorovich operators [20], q-Baskakov–Durrmeyer operators [19], q-Szász-beta operators [18], and q-Meyer–König–Zeller–Durrmeyer operators [15], has been constructed; see also [2]. In [6], Cai and Zeng defined q-gamma operators

$$ G_{n,q}(f;x)=\frac{[2n+3]! (q^{n+\frac{3}{2}}x )^{n+3}q ^{\frac{n(n+1)}{2}}}{[n]_{q}![n+2]_{q}!} \int _{0}^{\infty }{\frac{t ^{n}}{ (q^{n+\frac{3}{2}}x+t )_{q}^{2n+4}}f(t)}\,\mathrm{d} _{q} t,\quad x>0 $$
(2)

and gave their approximation properties.

Then many operators have been constructed with two parameters \((p,q)\)-integer based on post-quantum calculus (\((p,q)\)-calculus) which has been used efficiently in many areas of sciences such as Lie group, different equations, hypergeometric series, physical sciences, and so on. Recently, approximation by sequences of linear positive operators has been transferred to operators with \((p,q)\)-integer. Let us review some useful notations and definitions about \((p,q)\)-calculus in [2, 17, 21].

Let \(0< q< p\leq 1\). For each nonnegative integer n, the \((p,q)\)-integer \([n]_{p,q}\), \((p,q)\)-factorial \([n]_{p,q}!\) are defined by

$$ [n]_{p,q}=\frac{p^{n}-q^{n}}{p-q},\quad n=0,1,2,\ldots $$

and

$$ [n]_{p,q}!= \textstyle\begin{cases} [1]_{p,q}[2]_{p,q} \cdots [n]_{p,q},&n\geq 1; \\ 1,&n=0. \end{cases} $$

Further, the \((p,q)\)-power basis is defined by

$$ (x\oplus y)_{p,q}^{n}=(x+y) (px+qy) \bigl(p^{2}x+q^{2}y \bigr)\cdots \bigl(p^{n-1}x+q ^{n-1}y\bigr) $$

and

$$ (x\ominus y)_{p,q}^{n}=(x-y) (px-qy) \bigl(p^{2}x-q^{2}y \bigr)\cdots \bigl(p^{n-1}x-q ^{n-1}y\bigr). $$

Let n be a non-negative integer, the \((p,q)\)-gamma function is defined as

$$ \varGamma _{p,q}(n+1)=\frac{(p\ominus q)_{p,q}^{n}}{(p-q)^{n}}=[n]_{p,q}!,\quad 0< q< p \leq 1. $$

Aral and Gupta [1] proposed a \((p,q)\)-beta function of the second kind for \(m,n\in \mathbb{N}\) as follows:

$$ B_{p,q}(m,n)= \int _{0}^{\infty } \frac{x^{m-1}}{(1\oplus px)_{p,q}^{m+n}}\,\mathrm{d}_{p,q} x $$

and gave the relation of the \((p,q)\)-analogues of beta and gamma functions:

$$ B_{p,q}(m,n)=\frac{q\varGamma _{p,q}(m)\varGamma _{p,q}(n)}{(p^{m+1}q^{m-1})^{ \frac{m}{2}}\varGamma _{p,q}(m+n)}. $$

As a special case, if \(p=q=1\), \(B(m,n)=\frac{\varGamma (m)\varGamma (n)}{ \varGamma (m+n)}\). It is obvious that order is important for \((p,q)\)-setting, which is the reason why a \((p,q)\)-variant of beta function does not satisfy commutativity property, i.e., \(B_{p,q}(m,n) \neq B_{p,q}(n,m)\).

Let \(C_{B}[0,\infty )\) be the space of all real-valued continuous bounded functions f on the interval \([0,\infty )\) endowed with the norm

$$ \Vert f \Vert =\sup _{x\in [0,\infty )} \bigl\vert f(x) \bigr\vert . $$

Let \(\delta >0\) and \(C_{B}^{2}[0,\infty )=\{g:g',g''\in C_{B}[0, \infty )\}\), the following K-functional is defined:

$$ K(f;\delta )=\inf _{g\in C_{B}^{2}[0,\infty )}\bigl\{ \Vert f-g \Vert +\delta \bigl\Vert g'' \bigr\Vert \bigr\} . $$

Using DeVore–Lorentz theorem (see [10]), there exists a constant \(C>0\) such that

$$ K(f;\delta )\leq C\omega _{2} (f;\sqrt{\delta } ), $$
(3)

where

$$ \omega _{2}(f;\delta )=\sup _{0< \vert t \vert \leq \delta }\sup _{x\in [0,\infty )} \bigl\vert f(x+2t)-2f(x+t)+f(x) \bigr\vert $$

is the second order modulus of smoothness of f. Also, by \(\omega (f; \delta )\) we denote the usual modulus of continuity of \(f\in C_{B}[0, \infty )\) defined as

$$ \omega (f;\delta )=\sup _{0< \vert t \vert \leq \delta }\sup _{x\in [0,\infty )} \bigl\vert f(x+t)-f(x) \bigr\vert . $$

Let \(B_{x^{2}}[0,\infty )\) denote the function space of all functions f such that \(|f(x)|\leq C_{f}(1+x^{2})\), where \(C_{f}\) is a positive constant depending on f. By \(C_{x^{2}}[0,\infty )\) we denote the subspace of all continuous functions in the function space \(B_{x^{2}}[0, \infty )\). By \(C_{x^{2}}^{0}[0,\infty )\) we denote the subspace of all functions \(f\in C_{x^{2}}[0,\infty )\) for which \(\lim_{x\rightarrow \infty }\frac{|f(x)|}{1+x^{2}}\) is endowed with the norm

$$ \Vert f \Vert _{x^{2}}=\sup _{x\in [0,\infty )}\frac{ \vert f(x) \vert }{1+x^{2}}. $$

For \(a>0\), the modulus of continuity of f on \([0,a]\) is defined as follows:

$$ \omega _{a}(f;\delta )=\sup _{ \vert y-x \vert < \delta }\sup _{0\leq x,y\leq a} \bigl\vert f(y)-f(x) \bigr\vert . $$

As is known, if f is not uniformly continuous on \([0,\infty )\), we cannot get \(\omega (f;\delta )\rightarrow 0 \) as \(\delta \rightarrow 0\). In [38], Yuksel and Ispir defined the weighted modulus of continuity \(\varOmega (f;\delta )=\sup_{0< h\leq \delta ,x\geq 0} \frac{|f(x+h)-f(x)|}{1+(x+h)^{2}}\) while \(f\in C_{x^{2}}^{0}[0, \infty )\) and proved the properties of monotone increasing about \(\varOmega (f;\delta )\) as \(\delta >0\) and the inequality \(\varOmega (f; \lambda \delta )\leq (1+\lambda )\varOmega (f;\delta )\) while \(\lambda >0\) and \(f\in C_{x^{2}}^{0}[0,\infty )\).

Let \(f\in C_{B}[0,\infty )\), \(M>0\), and \(\gamma \in (0,1]\). We recall that \(f\in \mathrm{Lip}_{M}(\gamma )\) if the following inequality

$$ \bigl\vert f(x)-f(y) \bigr\vert \leq M \vert x-y \vert ^{\gamma }, \quad x,y\in [0,\infty ) $$

is satisfied. Let F be a subset of the interval \([0,\infty )\), we define that \(f\in \mathrm{Lip}_{M}(\gamma ,F)\) if the following inequality

$$ \bigl\vert f(x)-f(y) \bigr\vert \leq M \vert x-y \vert ^{\gamma }, \quad x\in F \mbox{ and } y\in [0,\infty ) $$

holds.

Recently, Mursaleen first applied \((p,q)\)-calculus in approximation theory and introduced the \((p,q)\)-analogue of Bernstein operators [33], \((p,q)\)-Bernstein–Stancu operators [34], \((p,q)\)-Bernstein–Schurer operators [35] and investigated their approximation properties. In addition, many well-known approximation operators with \((p,q)\)-integer, such as \((p,q)\)-Bernstein–Stancu–Schurer–Kantorovich operators [8], \((p,q)\)-Szász–Baskakov operators [16], \((p,q)\)-Baskakov-beta operators [30] have been introduced. All this achievement motivates us to construct the \((p,q)\)-analogue of the gamma operator (1), as we know that many researchers have studied approximation properties of the gamma operators and their modifications (see [7, 9, 24, 39]). The rest of the paper is organized as follows. In Sect. 2, we define the \((p,q)\)-gamma operators and obtain the moments and the central moments of them. In Sect. 3, we study the properties of the \((p,q)\)-gamma operators about Lipschitz condition. Then some direct theorems about local approximation, rate of convergence, weighted approximation, and Voronovskaja-type approximation are obtained.

\((p,q)\)-gamma operators and moments

We first define the analogue of gamma operators via \((p,q)\)-calculus as follows.

Definition 2.1

For \(n\in \mathbb{N}\), \(x\in (0,\infty )\) and \(0< q< p\leq 1\), the \((p,q)\)-gamma operators can be defined as follows:

$$ G_{n}^{p,q}(f;x)=\frac{x^{n+3} (q^{n+\frac{3}{2}} )^{n+3}p ^{n^{2}+\frac{7}{2}n+\frac{7}{2}}}{B_{p,q}(n+1,n+3)} \int _{0}^{\infty } \frac{t^{n}}{ ((pq)^{n+\frac{3}{2}}x\oplus t )_{p,q}^{2n+4}}f(t) \,\mathrm{d}_{p,q} t. $$

Operators \(G_{n}^{p,q}\) are linear and positive. For \(p=1\), they turn out to be the q-gamma operators defined in (2). We will derive the moments \(G_{n}^{p,q}(t^{k};x)\) and the central moments \(G_{n}^{p,q}((t-x)^{k};x)\) for \(k=0,1,2,3,4\).

Lemma 2.1

For \(x\in (0,\infty )\), \(0< q< p\leq 1\), and \(k=0,1,\ldots , n+2\), we have

$$ G_{n}^{p,q}\bigl(t^{k};x\bigr)= \frac{x^{k}(pq)^{k-\frac{k^{2}}{2}}[n+k]_{p,q}![n-k+2]_{p,q}!}{[n]_{p,q}![n+2]_{p,q}!}. $$
(4)

Proof

Using the properties of \((p,q)\)-beta function and \((p,q)\)-gamma function, we have

$$\begin{aligned} G_{n}^{p,q}\bigl(t^{k};x\bigr) =& \frac{x^{n+3} (q^{n+\frac{3}{2}} ) ^{n+3}p^{n^{2}+\frac{7}{2}n+\frac{7}{2}}}{B_{p,q}(n+1,n+3)} \int _{0} ^{\infty }\frac{t^{n+k}}{ ((pq)^{n+\frac{3}{2}}x\oplus t ) _{p,q}^{2n+4}} \,\mathrm{d}_{p,q} t \\ =&\frac{x^{n+3} (q^{n+\frac{3}{2}} )^{n+3}p^{n^{2}+ \frac{7}{2}n+\frac{7}{2}}}{B_{p,q}(n+1,n+3)} \int _{0}^{\infty }\frac{1}{(pq)^{(2n+3)(n+2)}x ^{2n+4}} \\ &{}\times \frac{t^{n+k}}{ (1\oplus \frac{pt}{xq^{n+\frac{3}{2}}p ^{n+\frac{5}{2}}} )_{p,q}^{2n+4}}\,\mathrm{d}_{p,q} t \\ =&\frac{x^{n+3} (q^{n+\frac{3}{2}} )^{n+3}p^{n^{2}+ \frac{7}{2}n+\frac{7}{2}}}{B_{p,q}(n+1,n+3)} \int _{0}^{\infty }\frac{ (xq^{n+\frac{3}{2}}p^{n+\frac{5}{2}} )^{n+k+1}}{(pq)^{(2n+3)(n+2)}x ^{2n+4}} \\ &{}\times \frac{ (\frac{t}{xq^{n+\frac{3}{2}}p^{n+\frac{5}{2}}} ) ^{n+k}}{ (1\oplus \frac{pt}{xq^{n+\frac{3}{2}}p^{n+\frac{5}{2}}} )_{p,q}^{2n+4}} \,\mathrm{d}_{p,q} \biggl(\frac{t}{xq^{n+\frac{3}{2}}p^{n+\frac{5}{2}}} \biggr) \\ =&\frac{x^{k}p^{kn+\frac{5}{2}k}q^{kn+\frac{3}{2}k}B_{p,q}(n+k+1,n-k+3)}{B _{p,q}(n+1,n+3)} \\ =& \frac{x^{k}(pq)^{k-\frac{k^{2}}{2}}[n+k]_{p,q}![n-k+2]_{p,q}!}{[n]_{p,q}![n+2]_{p,q}!}. \end{aligned}$$

Lemma 2.1 is proved. □

Lemma 2.2

For \(x\in (0,\infty )\), \(0< q< p\leq 1\), the following equalities hold:

1. \(G_{n}^{p,q}(1;x)=1\);

2. \(G_{n}^{p,q}(t;x)=\sqrt{\frac{p}{q}} (1- \frac{p^{n+1}}{[n+2]_{p,q}} )x\);

3. \(G_{n}^{p,q}(t^{2};x)=x^{2}\);

4. \(G_{n}^{p,q}(t^{3};x)= \frac{[n+3]_{p,q}x^{3}}{(pq)^{\frac{3}{2}}[n]_{p,q}}\);

5. \(G_{n}^{p,q}(t^{4};x)= \frac{[n+3]_{p,q}[n+4]_{p,q}x^{4}}{(pq)^{4}[n]_{p,q}[n-1]_{p,q}}\) for \(n>1\).

Proof

The proof of this lemma is an immediate consequence of Lemma 2.1. Hence the details are omitted. □

Lemma 2.3

Let \(n>1\) and \(x\in (0,\infty )\), then for \(0< q< p\leq 1\), we have the central moments as follows:

  1. 1.

    \(A(x):=G_{n}^{p,q}(t-x;x)= ( (\sqrt{\frac{p}{q}}-1 )-\sqrt{ \frac{p}{q}}\frac{p^{n+1}}{[n+2]_{p,q}} )x\);

  2. 2.

    \(B(x):=G_{n}^{p,q}((t-x)^{2};x)=-2 ( (\sqrt{ \frac{p}{q}}-1 )-\sqrt{\frac{p}{q}} \frac{p^{n+1}}{[n+2]_{p,q}} )x^{2}\);

  3. 3.

    \(G_{n}^{p,q}((t-x)^{4};x)= (\frac{[n+2]_{p,q}[n+3]_{p,q}[n+4]_{p,q}-4(pq)^{ \frac{5}{2}}[n-1]_{p,q}[n+2]_{p,q}[n+3]_{p,q}}{(pq)^{4}[n-1]_{p,q}[n]_{p,q}[n+2]_{p,q}}+ \frac{-4(pq)^{\frac{9}{2}}[n-1]_{p,q}[n]_{p,q}[n+1]_{p,q}+7(pq)^{4}[n-1]_{p,q}[n]_{p,q}[n+2]_{p,q}}{(pq)^{4}[n-1]_{p,q}[n]_{p,q}[n+2]_{p,q}} )x^{4}\).

Proof

Because \(G_{n}^{p,q}(t-x;x)=G_{n}^{p,q}(t;x)-x\), \(G_{n}^{p,q}((t-x)^{2};x)=G _{n}^{p,q}(t^{2};x)-2xG_{n}^{p,q}(t;x)+x^{2}\), and \(G_{n}^{p,q}((t-x)^{4};x)=G _{n}^{p,q}(t^{4};x)-4xG_{n}^{p,q}(t^{3};x)+6x^{2}G_{n}^{p,q}(t^{2};x)-4x ^{3}G_{n}^{p,q}(t;x)+x^{4}\), and from Lemma 2.2, we obtain Lemma 2.3 easily. □

Lemma 2.4

The sequences \((p_{n})\), \((q_{n})\) satisfy \(0< q_{n}< p_{n}\leq 1\) such that \(p_{n}\rightarrow 1\), \(q_{n}\rightarrow 1\) and \(p_{n}^{n}\rightarrow \alpha \), \(q_{n}^{n}\rightarrow \beta \), \([n]_{p_{n},q_{n}}\rightarrow \infty \) as \(n\rightarrow \infty \), then

$$\begin{aligned}& \lim _{n\rightarrow \infty }[n-1]_{p_{n},q_{n}}G_{n}^{p_{n},q _{n}}(t-x;x)=- \frac{\alpha +\beta }{2}x; \end{aligned}$$
(5)
$$\begin{aligned}& \lim _{n\rightarrow \infty }[n-1]_{p_{n},q_{n}}G_{n}^{p_{n},q _{n}} \bigl((t-x)^{2};x\bigr)=(\alpha +\beta )x^{2}; \end{aligned}$$
(6)
$$\begin{aligned}& \lim _{n\rightarrow \infty }[n-1]_{p_{n},q_{n}}G_{n}^{p_{n},q _{n}} \bigl((t-x)^{4};x\bigr)=0. \end{aligned}$$
(7)

Proof

Using

$$\begin{aligned} &\lim _{n\rightarrow \infty }[n-1]_{p_{n},q_{n}} \biggl( \biggl(\sqrt{\frac{p _{n}}{q_{n}}}-1 \biggr)-\sqrt{\frac{p_{n}}{q_{n}}}\frac{p_{n}^{n+1}}{[n+2]_{p _{n},q_{n}}} \biggr) \\ &\quad =\lim _{n\rightarrow \infty }[n+2]_{p_{n},q_{n}} \biggl( \biggl(\sqrt{\frac{p _{n}}{q_{n}}}-1 \biggr)-\sqrt{\frac{p_{n}}{q_{n}}}\frac{p_{n}^{n+1}}{[n+2]_{p _{n},q_{n}}} \biggr) \\ &\quad =\lim _{n\rightarrow \infty } \biggl(\frac{p_{n}^{n+2}-q_{n} ^{n+2}}{p_{n}-q_{n}}\frac{\sqrt{p_{n}}-\sqrt{q_{n}}}{\sqrt{q _{n}}}-\sqrt{ \frac{p_{n}}{q_{n}}}p_{n}^{n+1} \biggr) \\ &\quad =\frac{\alpha -\beta }{2}-\alpha =-\frac{\alpha +\beta }{2}, \end{aligned}$$

we get (5) and (6) easily. Let \(k=n-2\), we have

$$\begin{aligned} &[n+2]_{p_{n},q_{n}}[n+3]_{p_{n},q_{n}}[n+4]_{p_{n},q_{n}} \\ &\quad =\bigl(q_{n}^{3}[k]_{p_{n},q_{n}}+p_{n}^{k}[3]_{p_{n},q_{n}} \bigr) \bigl(q_{n}^{4}[k]_{p _{n},q_{n}}+p_{n}^{k}[4]_{p_{n},q_{n}} \bigr) \bigl(q_{n}^{5}[k]_{p_{n},q_{n}}+p _{n}^{k}[5]_{p_{n},q_{n}}\bigr) \\ &\quad \sim q_{n}^{12}[k]_{p_{n},q_{n}}^{3}+p_{n}^{k} \bigl(q_{n}^{7}[5]_{p _{n},q_{n}}+q_{n}^{8}[4]_{p_{n},q_{n}}+q_{n}^{9}[3]_{p_{n},q_{n}} \bigr)[k]_{p _{n},q_{n}}^{2}. \end{aligned}$$

Similarly, we can obtain

$$\begin{aligned}& [n-1]_{p_{n},q_{n}}[n+2]_{p_{n},q_{n}}[n+3]_{p_{n},q_{n}}\sim q_{n} ^{7}[k]_{p_{n},q_{n}}^{3}+p^{k}_{n} \bigl(q_{n}^{3}[4]_{p_{n},q_{n}}+q_{n} ^{4}[3]_{p_{n},q_{n}}\bigr)[k]_{p_{n},q_{n}}^{2}, \\& [n-1]_{p_{n},q_{n}}[n]_{p_{n},q_{n}}[n+2]_{p_{n},q_{n}}\sim q_{n}^{4}[k]_{p _{n},q_{n}}^{3}+p^{k}_{n} \bigl(q_{n}^{3}+q_{n}[3]_{p_{n},q_{n}} \bigr)[k]_{p_{n},q _{n}}^{2}, \\& [n-1]_{p_{n},q_{n}}[n]_{p_{n},q_{n}}[n+1]_{p_{n},q_{n}}\sim q_{n}^{3}[k]_{p _{n},q_{n}}^{3}+p^{k}_{n} \bigl(q_{n}^{2}+q_{n}[2]_{p_{n},q_{n}} \bigr)[k]_{p_{n},q _{n}}^{2}. \end{aligned}$$

By Lemma 2.3, we can have

$$ G_{n}^{p_{n},q_{n}}\bigl((t-x)^{4};x\bigr)\sim \biggl(A_{n}+\frac{1}{[k]_{p_{n},q _{n}}}B_{n} \biggr)x^{4}, $$

where \(A_{n}=q_{n}^{12}-4p_{n}^{\frac{5}{2}}q_{n}^{\frac{19}{2}}-4p _{n}^{\frac{9}{2}}q_{n}^{\frac{15}{2}}+7p_{n}^{4}q_{n}^{8}\) and

$$\begin{aligned} B_{n}= {}&p_{n}^{k} \bigl(q_{n}^{7}[5]_{p_{n},q_{n}}+q_{n}^{8}[4]_{p _{n},q_{n}}+q_{n}^{9}[3]_{p_{n},q_{n}}-4(p_{n}q_{n})^{\frac{5}{2}} \bigl(q _{n}^{3}[4]_{p_{n},q_{n}}+q_{n}^{4}[3]_{p_{n},q_{n}} \bigr) \\ &{} -4(p_{n}q_{n})^{\frac{9}{2}}\bigl(q_{n}^{2}+q_{n}[2]_{p_{n},q_{n}} \bigr)+7(p _{n}q_{n})^{4}\bigl(q_{n}^{3}+q_{n}[3]_{p_{n},q_{n}} \bigr) \bigr). \end{aligned}$$

Set \(P=\sqrt{p_{n}}\), \(Q=\sqrt{q_{n}}\), by

$$\begin{aligned} A_{n} &= P^{24}-4P^{5}Q^{19}-4P^{9}Q^{15}+7P^{8}Q^{16} \\ &\sim P^{9}-4P^{5}Q^{4}-4P^{9}+7P^{8}Q \\ &=3P^{5}\bigl(P^{4}-Q^{4}\bigr)-Q^{4} \bigl(P^{5}-Q^{5}\bigr)-7P^{8}(P-Q) \\ &=(P-Q) \Biggl(3P^{5}\sum _{i=0}^{3}P^{i}Q^{3-i}-Q^{4} \sum _{i=0}^{4}P^{i}Q^{4-i}-7P^{8} \Biggr), \end{aligned}$$

we easily obtain

$$\begin{aligned}{} [n-1]_{p_{n},q_{n}}A_{n} &\sim [n]_{p_{n},q_{n}}(P-Q) \Biggl(3P^{5} \sum _{i=0}^{3}P^{i}Q^{3-i}-Q^{4} \sum _{i=0}^{4}P^{i}Q ^{4-i}-7P^{8} \Biggr) \\ &\sim \frac{p_{n}^{n}-q_{n}^{n}}{p_{n}-q_{n}}\frac{p_{n}-q_{n}}{\sqrt{p _{n}}+\sqrt{q_{n}}} \Biggl(3P^{5}\sum _{i=0}^{3}P^{i}Q^{3-i}-Q ^{4} \sum _{i=0}^{4}P^{i}Q^{4-i}-7P^{8} \Biggr) \\ &\sim \frac{a-b}{2}(3\times 4-5-7)=0. \end{aligned}$$

Similarly, \(B_{n}\sim 5+4+3-4\times (4+3)-4\times (1+2)+7\times (1+3)=0\), we obtain (7). □

Approximation properties of \((p,q)\)-gamma operators

In this section, we research the approximation properties of \((p,q)\)-gamma operators. The following two theorems show approximation properties about Lipschitz functions.

Theorem 3.1

Let \(0< q< p\leq 1\) and F be any bounded subset of the interval \([0,\infty )\). If \(f\in C_{B}[0,\infty )\cap \mathrm{Lip}_{M}( \gamma , F)\), then, for all \(x\in (0,\infty )\), we have

$$ \bigl\vert G_{n}^{p,q}(f;x)-f(x) \bigr\vert \leq M \bigl( \bigl(B(x)\bigr)^{\frac{\gamma }{2}}+2d^{ \gamma }(x;F) \bigr), $$

where \(d(x;F)\) is the distance between x and F defined by \(d(x;F)=\inf \{|x-y|:y\in F\}\).

Proof

Let be the closure of F in \([0,\infty )\). Using the properties of infimum, there is at least a point \(y_{0}\in \overline{F}\) such that \(d(x;F)=|x-y_{0}|\). By the triangle inequality, we can obtain

$$\begin{aligned} \bigl\vert G_{n}^{p,q}(f;x)-f(x) \bigr\vert &\leq G_{n}^{p,q}\bigl( \bigl\vert f(x)-f(t) \bigr\vert ;x\bigr) \\ &\leq G_{n}^{p,q}\bigl( \bigl\vert f(x)-f(y_{0}) \bigr\vert ;x\bigr)+G_{n}^{p,q}\bigl( \bigl\vert f(t)-f(y_{0}) \bigr\vert ;x\bigr) \\ &\leq M \bigl(G_{n}^{p,q}\bigl( \vert t-y_{0} \vert ^{\gamma };x\bigr)+G_{n}^{p,q}\bigl( \vert x-y _{0} \vert ^{\gamma };x\bigr) \bigr) \\ &\leq M \bigl(G_{n}^{p,q}\bigl( \vert x-t \vert ^{\gamma };x\bigr)+2d^{\gamma }(x;F) \bigr). \end{aligned}$$

Choosing \(k_{1}=\frac{2}{\gamma }\) and \(k_{2}=\frac{2}{2-\gamma }\) and using the well-known Hölder inequality, we have

$$\begin{aligned} \bigl\vert G_{n}^{p,q}(f;x)-f(x) \bigr\vert &\leq M \bigl( \bigl(G_{n}^{p,q}\bigl( \vert x-t \vert ^{k_{1} \gamma };x\bigr) \bigr)^{\frac{1}{k_{1}}} \bigl(G_{n}^{p,q} \bigl(1^{k_{2}};x\bigr) \bigr) ^{\frac{1}{k_{2}}}+2d^{\gamma }(x;F) \bigr) \\ &\leq M \bigl(G_{n}^{p,q}\bigl((x-t)^{2};x \bigr)^{\frac{\gamma }{2}}+2d^{ \gamma }(x;F) \bigr) \\ &=M \bigl(\bigl(B(x)\bigr)^{\frac{\gamma }{2}}+2d^{\gamma }(x;F) \bigr). \end{aligned}$$

This completes the proof. □

Theorem 3.2

Let \(0< q< p\leq 1\). Then, for all \(f\in \mathrm{Lip}_{M}(\gamma )\), we have

$$ \bigl\vert G_{n}^{p,q}(f;x)-f(x) \bigr\vert \leq MB^{\frac{\gamma }{2}}(x). $$

Proof

Using the monotonicity of the operators \(G_{n}^{p,q}\) and the Hölder inequality, we can obtain

$$\begin{aligned} \bigl\vert G_{n}^{p,q}(f;x)-f(x) \bigr\vert &\leq G_{n}^{p,q} \bigl( \bigl\vert f(t)-f(x) \bigr\vert ;x \bigr) \leq MG_{n}^{p,q} \bigl( \vert t-x \vert ^{\gamma };x \bigr) \\ &=MG_{n}^{p,q} \bigl(\bigl( \vert t-x \vert ^{2}\bigr)^{\frac{\gamma }{2}};x \bigr) \leq M \bigl(G_{n}^{p,q} \bigl((t-x)^{2};x\bigr) \bigr)^{\frac{\gamma }{2}}=MB ^{\frac{\gamma }{2}}(x). \end{aligned}$$

 □

The third theorem is a direct local approximation theorem for the operators \(G_{n}^{p,q}(f;x)\).

Theorem 3.3

Let \(0< q< p\leq 1\), \(f\in C_{B}[0,\infty )\). Then, for every \(x\in (0,\infty )\), there exists a positive constant \(C_{1}\) such that

$$ \bigl\vert G_{n}^{p,q}(f;x)-f(x) \bigr\vert \leq C_{1}\omega _{2} \bigl(f;\sqrt{B(x)+A ^{2}(x)} \bigr)+\omega \bigl(f; \bigl\vert A(x) \bigr\vert \bigr). $$

Proof

For \(x\in (0,\infty )\), we consider new operators \(H_{n}^{p,q}(f;x)\) defined by

$$ H_{n}^{p,q}(f;x)=G_{n}^{p,q}(f;x)+f(x)-f \bigl(A(x)+x \bigr). $$

Using the operator above and Lemma 2.3, we have

$$ H_{n}^{p,q}(t-x;x)= G_{n}^{p,q}(t-x;x)-A(x)=0 . $$

Let \(x,t\in (0,\infty )\) and \(g\in C_{B}^{2}[0,\infty )\). Using Taylor’s expansion, we can obtain

$$ g(t)=g(x)+g'(x) (t-x)+ \int _{x}^{t}g''(u) (t-u) \,\mathrm{d} u. $$

Hence,

$$\begin{aligned} \bigl\vert H_{n}^{p,q}(g;x)-g(x) \bigr\vert & = \biggl\vert g'(x)H_{n}^{p,q} \bigl((t-x);x \bigr)+H _{n}^{p,q} \biggl( \int _{x}^{t}g''(u) (t-u) \,\mathrm{d} u;x \biggr) \biggr\vert \\ &\leq \biggl\vert H_{n}^{p,q} \biggl( \int _{x}^{t}g''(u) (t-u) \,\mathrm{d} u;x \biggr) \biggr\vert \\ &\leq \biggl\vert G_{n}^{p,q} \biggl( \int _{x}^{t}g''(u) (t-u) \,\mathrm{d} u;x \biggr) - \int _{x}^{A(x)+x}g''(u) \bigl(A(x)+x-u\bigr)\,\mathrm{d} u \biggr\vert \\ &\leq G_{n}^{p,q} \biggl( \int _{x}^{t} \bigl\vert g''(u) \bigr\vert (t-u)\,\mathrm{d} u;x \biggr) + \biggl\vert \int _{x}^{A(x)+x} \bigl\vert g''(u) \bigr\vert \bigl(A(x)+x-u\bigr)\,\mathrm{d} u \biggr\vert \\ &\leq \bigl(B(x)+A^{2}(x)\bigr) \bigl\Vert g'' \bigr\Vert . \end{aligned}$$

Using \(|G_{n}^{p,q}(f;x)|\leq \|f\|\), we have

$$\begin{aligned} &\bigl\vert G_{n}^{p,q}(f;x)-f(x) \bigr\vert \\ &\quad = \bigl\vert H_{n}^{p,q}(f;x)+f \bigl(A(x)+x \bigr)-2f(x) \bigr\vert \\ &\quad \leq \bigl\vert H_{n}^{p,q}(f-g;x)-(f-g) (x) \bigr\vert + \bigl\vert H_{n}^{p,q}(g;x)-g(x) \bigr\vert + \bigl\vert f \bigl(A(x)+x \bigr)-f(x) \bigr\vert \\ &\quad \leq 4 \Vert f-g \Vert +\bigl(B(x)+A^{2}(x)\bigr) \bigl\Vert g'' \bigr\Vert +\omega \bigl(f; \bigl\vert A(x) \bigr\vert \bigr). \end{aligned}$$

Taking infimum over all \(g\in C_{B}^{2}[0,\infty )\) and using (3), we can obtain the desired assertion. □

The fourth theorem is a result about the rate of convergence for the operators \(G_{n}^{p,q}(f;x)\):

Theorem 3.4

Let \(f\in C_{x^{2}}[0,\infty )\), \(0< q< p\leq 1\), and \(a>0\), we have

$$ \bigl\Vert G_{n}^{p,q}(f;x)-f(x) \bigr\Vert _{C(0,a]}\leq 4C_{f}\bigl(1+a^{2}\bigr)B(a)+2\omega _{a+1}\bigl(f;\sqrt{B(a)}\bigr). $$

Proof

For all \(x\in (0,a]\) and \(t>a+1\), we easily have \((t-x)^{2}\geq (t-a)^{2} \geq 1\), therefore,

$$\begin{aligned} \begin{aligned} \bigl\vert f(t)-f(x) \bigr\vert & \leq \bigl\vert f(t) \bigr\vert + \bigl\vert f(x) \bigr\vert \leq C_{f}\bigl(2+x^{2}+t^{2}\bigr) \\ &=C_{f} \bigl(2+x^{2}+(x-t-x)^{2} \bigr)\leq C_{f} \bigl(2+3x^{2}+2(x-t)^{2} \bigr) \\ &\leq C_{f}\bigl(4+3x^{2}\bigr) (t-x)^{2}\leq 4C_{f}\bigl(1+a^{2}\bigr) (t-x)^{2}, \end{aligned} \end{aligned}$$
(8)

and for all \(x\in (0,a]\), \(t\in (0,a+1]\), and \(\delta >0\), we have

$$ \bigl\vert f(t)-f(x) \bigr\vert \leq \omega _{a+1} \bigl(f, \vert t-x \vert \bigr)\leq \biggl(1+\frac{ \vert t-x \vert }{ \delta } \biggr)\omega _{a+1}(f;\delta ). $$
(9)

From (8) and (9), we get

$$ \bigl\vert f(t)-f(x) \bigr\vert \leq 4C_{f}\bigl(1+a^{2} \bigr) (t-x)^{2}+ \biggl(1+\frac{ \vert t-x \vert }{ \delta } \biggr)\omega _{a+1}(f;\delta ). $$

By Schwarz’s inequality and Lemma 2.3, we have

$$ \begin{aligned} &\bigl\vert G_{n}^{p,q}(f;x)-f(x) \bigr\vert \\ &\quad \leq G_{n}^{p,q}\bigl( \bigl\vert f(t)-f(x) \bigr\vert ;x\bigr) \\ &\quad \leq 4C _{f}\bigl(1+a^{2}\bigr)G_{n}^{p,q} \bigl((t-x)^{2};x\bigr)+G_{n}^{p,q} \biggl( \biggl(1+ \frac{ \vert t-x \vert }{ \delta } \biggr);x \biggr)\omega _{a+1}(f;\delta ) \\ &\quad \leq 4C_{f}\bigl(1+a ^{2}\bigr)G_{n}^{p,q} \bigl((t-x)^{2};x\bigr)+\omega _{a+1}(f;\delta ) \biggl(1+ \frac{1}{ \delta }\sqrt{G_{n}^{p,q} \bigl((t-x)^{2};x\bigr)} \biggr) \\ &\quad \leq 4C_{f}\bigl(1+a ^{2}\bigr)B(x)+\omega _{a+1}(f;\delta ) \biggl(1+\frac{1}{\delta } \sqrt{B(x)} \biggr) \\ &\quad \leq 4C_{f}\bigl(1+a^{2}\bigr)B(a)+\omega _{a+1}(f;\delta ) \biggl(1+\frac{1}{\delta }\sqrt{B(a)} \biggr). \end{aligned} $$

By taking \(\delta =\sqrt{B(a)}\) and supremum over all \(x\in (0,a]\), we accomplish the proof of Theorem 3.4. □

The following three results are theorems about weighted approximation for the operators \(G_{n}^{p,q}(f;x)\).

Theorem 3.5

Let \(f\in C^{0}_{x^{2}}[0,\infty )\) and the sequences \((p_{n})\), \((q_{n})\) satisfy \(0< q_{n}< p_{n}\leq 1\) such that \(p_{n}^{n}\rightarrow 1\), \(q_{n}^{n}\rightarrow 1\), \([n]_{p_{n},q_{n}}\rightarrow \infty \) as \(n\rightarrow \infty \), then there exists a positive integer \(N\in \mathbb{N_{+}}\) such that, for all \(n>N\) and \(\nu >0\), the inequality

$$ \sup _{x\in (0,\infty )}\frac{ \vert G_{n}^{p_{n},q_{n}}(f;x)-f(x) \vert }{(1+x ^{2})^{\frac{3}{2}+\nu }}\leq 4\sqrt{2}\varOmega \biggl(f;\frac{1}{\sqrt{[n-1]_{p _{n},q_{n}}}} \biggr) $$
(10)

holds.

Proof

For \(t>0\), \(x\in (0,\infty )\) and \(\delta >0\), by the definition and properties of \(\varOmega (f;\delta )\), we get

$$\begin{aligned} \bigl\vert f(t)-f(x) \bigr\vert &\leq \bigl(1+ \bigl(x+ \vert x-t \vert \bigr) \bigr)^{2}\varOmega \bigl(f; \vert t-x \vert \bigr) \\ &\leq 2\bigl(1+x^{2}\bigr) \bigl(1+(t-x)^{2} \bigr) \biggl(1+\frac{ \vert t-x \vert }{ \delta } \biggr)\varOmega (f;\delta ). \end{aligned}$$

Using \(p_{n}^{n}\rightarrow 1\), \(q_{n}^{n}\rightarrow 1\), \([n]_{p_{n},q _{n}}\rightarrow \infty \) as \(n\rightarrow \infty \) and Lemma 2.4, there exists a positive integer \(N\in \mathbb{N_{+}}\) such that, for all \(n>N\),

$$\begin{aligned}& G_{n}^{p_{n},q_{n}}\bigl((t-x)^{2};x\bigr) \leq \frac{2(1+x^{2})}{[n-1]_{p_{n},q _{n}}}, \end{aligned}$$
(11)
$$\begin{aligned}& G_{n}^{p_{n},q_{n}}\bigl((t-x)^{4};x\bigr) \leq 1. \end{aligned}$$
(12)

Since \(G_{n}^{p_{n},q_{n}}\) is linear and positive, we have

$$\begin{aligned} \begin{aligned}[b] \bigl\vert G_{n}^{p_{n},q_{n}}(f;x)-f(x) \bigr\vert \leq {}&2\bigl(1+x^{2}\bigr)\varOmega (f; \delta ) \biggl\{ 1+G_{n}^{p_{n},q_{n}} \bigl((t-x)^{2};x \bigr) \\ &{}+G_{n}^{p_{n},q_{n}} \biggl( \bigl(1+(t-x)^{2} \bigr) \frac{ \vert t-x \vert }{ \delta };x \biggr) \biggr\} . \end{aligned} \end{aligned}$$
(13)

To estimate the second term of (13), applying the Cauchy–Schwarz inequality and \((x+y)^{2}\leq 2(x^{2}+y^{2})\), we have

$$ G_{n}^{p_{n},q_{n}} \biggl( \bigl(1+(t-x)^{2} \bigr) \frac{ \vert t-x \vert }{ \delta };x \biggr)\leq \sqrt{2} \bigl(G_{n}^{p_{n},q_{n}} \bigl(1+(t-x)^{4};x \bigr) \bigr) ^{\frac{1}{2}} \biggl(G_{n}^{p_{n},q_{n}} \biggl(\frac{(t-x)^{2}}{ \delta ^{2}};x \biggr) \biggr)^{\frac{1}{2}}. $$

By (11) and (12),

$$ G_{n}^{p_{n},q_{n}} \biggl( \bigl(1+(t-x)^{2} \bigr) \frac{ \vert t-x \vert }{ \delta };x \biggr)\leq \frac{2\sqrt{2}(1+x^{2})^{\frac{1}{2}}}{ \delta [n-1]_{p_{n},q_{n}}}. $$

Taking \(\delta =\frac{1}{\sqrt{[n-1]_{p_{n},q_{n}}}}\), we can obtain

$$ \bigl\vert G_{n}^{p_{n},q_{n}}(f;x)-f(x) \bigr\vert \leq 4 \sqrt{2}\bigl(1+x^{2}\bigr)^{ \frac{3}{2}}\varOmega \biggl(f; \frac{1}{\sqrt{[n-1]_{p_{n},q_{n}}}} \biggr). $$

The proof is completed. □

Theorem 3.6

Let the sequences \((p_{n})\), \((q_{n})\) satisfy \(0< q_{n}< p_{n}\leq 1\) such that \(p_{n}\rightarrow 1\), \(q_{n}\rightarrow 1\), and \(p_{n}^{n} \rightarrow \alpha \), \(q_{n}^{n}\rightarrow \beta \), \([n]_{p_{n},q _{n}}\rightarrow \infty \) as \(n\rightarrow \infty \). Then, for \(f\in C^{0}_{x^{2}}[0,\infty )\), we have

$$ \lim _{n\rightarrow \infty } \bigl\Vert G_{n}^{p_{n},q_{n}}(f;x)-f(x) \bigr\Vert _{x^{2}}=0. $$
(14)

Proof

By the Korovkin theorem in [14], we see that it is sufficient to verify the following three conditions:

$$ \lim _{n\rightarrow \infty } \bigl\Vert G_{n}^{p_{n},q_{n}} \bigl(t^{k};x\bigr)-x ^{k} \bigr\Vert _{x^{2}}=0,\quad k=0,1,2. $$
(15)

Since \(G_{n}^{p_{n},q_{n}}(1;x)=1\), \(G_{n}^{p_{n},q_{n}}(t^{2};x)=x ^{2}\), then (15) holds true for \(k=0,2\). By Lemma 2.2, we can get

$$\begin{aligned} \bigl\Vert G_{n}^{p_{n},q_{n}}(t;x)-x \bigr\Vert _{x^{2}} &=\sup _{x\in (0,\infty )}\frac{1}{1+x ^{2}} \bigl\vert G_{n}^{p_{n},q_{n}}(t;x)-x \bigr\vert \\ &=\sup _{x\in (0,\infty )}\frac{x}{1+x^{2}} \biggl\vert \frac{\sqrt{p _{n}}-\sqrt{q_{n}}}{\sqrt{q_{n}}}- \sqrt{\frac{p_{n}}{q_{n}}}\frac{p _{n}^{n+1}}{[n+2]_{p_{n},q_{n}}} \biggr\vert \\ &\leq \sup _{x\in (0,\infty )} \biggl\vert \frac{\sqrt{p_{n}}-\sqrt{q _{n}}}{\sqrt{q_{n}}}-\sqrt{ \frac{p_{n}}{q_{n}}}\frac{p_{n}^{n+1}}{[n+2]_{p _{n},q_{n}}} \biggr\vert \rightarrow 0,\quad n \rightarrow \infty . \end{aligned}$$

Thus the proof is completed. □

Theorem 3.7

Let the sequences \((p_{n})\), \((q_{n})\) satisfy \(0< q_{n}< p_{n}\leq 1\) such that \(p_{n}\rightarrow 1\), \(q_{n}\rightarrow 1\), \([n]_{p_{n},q _{n}}\rightarrow \infty \) as \(n\rightarrow \infty \). For every \(f\in C_{x^{2}}[0,\infty )\) and \(\kappa >0\), we have

$$ \lim _{n\rightarrow \infty }\sup _{x\in (0,\infty )}\frac{ \vert G _{n}^{p_{n},q_{n}}(f;x)-f(x) \vert }{(1+x^{2})^{1+\kappa }}=0. $$

Proof

Let \(x_{0}\in (0,\infty )\) be arbitrary but fixed. Then

$$\begin{aligned} \begin{aligned}[b] \sup _{x\in (0,\infty )}\frac{ \vert G_{n}^{p_{n},q_{n}}(f;x)-f(x) \vert }{(1+x ^{2})^{1+\kappa }} \leq{} &\sup _{x\in (0,x_{0}]}\frac{ \vert G_{n}^{p _{n},q_{n}}(f;x)-f(x) \vert }{(1+x^{2})^{1+\kappa }} \\ &{}+ \sup _{x\in (x_{0},\infty )}\frac{ \vert G_{n}^{p_{n},q_{n}}(f;x)-f(x) \vert }{(1+x ^{2})^{1+\kappa }} \\ \leq{}& \bigl\Vert G_{n}^{p_{n},q_{n}}(f;x)-f(x) \bigr\Vert _{C(0,x_{0}]} \\ &{}+C_{f}\sup _{x\in (x_{0},\infty )}\frac{ \vert G_{n}^{p_{n},q_{n}}((1+t ^{2});x) \vert }{(1+x^{2})^{1+\kappa }} \\ &{}+\sup _{x\in (x_{0},\infty )}\frac{ \vert f(x) \vert }{(1+x^{2})^{1+ \kappa }}. \end{aligned} \end{aligned}$$
(16)

Since \(|f(x)|\leq C_{f}(1+x^{2})\), we have \(\sup_{x\in (x_{0},\infty )}\frac{|f(x)|}{(1+x^{2})^{1+\kappa }} \leq \frac{C_{f}}{(1+x_{0}^{2})^{\kappa }}\). Let \(\epsilon >0\) be arbitrary. We can choose \(x_{0}\) to be so large that

$$ \frac{C_{f}}{(1+x_{0}^{2})^{\kappa }}< \epsilon . $$
(17)

In view of Lemma 2.2, while \(x\in (x_{0},\infty )\), we obtain

$$ C_{f}\lim _{n\rightarrow \infty }\frac{ \vert G_{n}^{p_{n},q_{n}}((1+t ^{2});x) \vert }{(1+x^{2})^{1+\kappa }} =C_{f} \frac{(1+x^{2})}{(1+x^{2})^{1+ \kappa }}=\frac{C_{f}}{(1+x^{2})^{\kappa }}\leq \frac{C_{f}}{(1+x_{0} ^{2})^{\kappa }}< \epsilon . $$

Using Theorem 3.4, we can see that the first term of inequality (16) implies that

$$ \bigl\Vert G_{n}^{p_{n},q_{n}}(f;x)-f(x) \bigr\Vert _{C(0,x_{0}]}< \epsilon ,\quad \mbox{as } n\rightarrow \infty . $$
(18)

Combining (16)–(18), we get the desired result. □

The last result is a Voronovskaja-type asymptotic formula for the operators \(G_{n}^{p,q}(f;x)\).

Theorem 3.8

Let \(f\in C_{B}^{2}[0,\infty )\) and the sequences \((p_{n})\), \((q_{n})\) satisfy \(0< q_{n}< p_{n}\leq 1\) such that \(p_{n}\rightarrow 1\), \(q_{n}\rightarrow 1\) and \(p_{n}^{n}\rightarrow \alpha \), \(q_{n}^{n} \rightarrow \beta \), \([n]_{p_{n},q_{n}}\rightarrow \infty \) as \(n\rightarrow \infty \), where \(0\leq \alpha ,\beta <1\). Then, for all \(x\in (0,\infty )\),

$$ \lim _{n\rightarrow \infty }[n-1]_{p_{n},q_{n}} \bigl(G_{n}^{p _{n},q_{n}}(f;x)-f(x) \bigr)=\frac{\alpha +\beta }{2} \bigl(-xf'(x)+x ^{2}f''(x) \bigr). $$
(19)

Proof

Let \(x\in (0,\infty )\) be fixed. By Taylor’s expansion formula, we obtain

$$ f(t)=f(x)+f'(x) (t-x)+ \biggl(\frac{1}{2}f''(x)+ \varTheta _{p_{n},q_{n}}(t,x) \biggr) (t-x)^{2}, $$

where \(\varTheta _{p_{n},q_{n}}(x,t)\) is bounded and \(\lim_{t\rightarrow x}\varTheta _{p_{n},q_{n}}(t,x)=0\). By applying the operator \(G_{n}^{p_{n},q_{n}}(f;x)\) to the relation above, we obtain

$$\begin{aligned} G_{n}^{p_{n},q_{n}}(f;x)-f(x)={}&f'(x)G_{n}^{p_{n},q_{n}} \bigl((t-x);x \bigr)+ \frac{1}{2}f''(x)G_{n}^{p_{n},q_{n}} \bigl((t-x)^{2};x \bigr) \\ &{}+G_{n}^{p_{n},q_{n}} \bigl(\varTheta _{p_{n},q_{n}}(t,x) (t-x)^{2};x \bigr). \end{aligned}$$

Since \(\lim_{t\rightarrow x}\varTheta _{p_{n},q_{n}}(t,x)=0\), then for all \(\epsilon >0\), there exists a positive constant \(\delta >0\) which implies \(|\varTheta _{p_{n},q_{n}}(t,x)|<\epsilon \) for all fixed \(x\in (0,\infty )\), where n is large enough, while \(|t-x|\leq \delta \), then \(|\varTheta _{p_{n},q_{n}}(t,x)|<\frac{C_{2}}{\delta ^{2}}(t-x)^{2}\), where \(C_{2}\) is a positive constant. Using Lemma 2.4, we obtain

$$\begin{aligned} &[n-1]_{p_{n},q_{n}} \bigl\vert G_{n}^{p_{n},q_{n}} \bigl(\varTheta (t,x) (t-x)^{2};x \bigr) \bigr\vert \\ &\quad \leq \epsilon [n-1]_{p_{n},q_{n}}G_{n}^{p_{n},q_{n}} \bigl((t-x)^{2};x \bigr) \\ &\qquad {}+\frac{C_{2}}{\delta ^{2}}[n-1]_{p_{n},q_{n}}G_{n}^{p_{n},q_{n}} \bigl((t-x)^{4};x \bigr) \rightarrow 0\quad (n\rightarrow \infty ). \end{aligned}$$

The proof is completed. □

References

  1. 1.

    Aral, A., Gupta, V.: \((p,q)\)-type beta function of second kind. Adv. Oper. Theor. 1(1), 134–146 (2016)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Aral, A., Gupta, V., Agarwal, R.P.: Application of q-Calculus in Operator Theory. Springer, Berlin (2013)

    Google Scholar 

  3. 3.

    Baskakov, V.A.: Primer posledovatel’nosti lineinyh polozitel’nyh operatorov v prostranstve neprerivnyh funkeil (An example of a sequence of linear positive operators in the space of continuous functions). Dokl. Akad. Nauk SSSR 113, 249–251 (1957)

    MathSciNet  Google Scholar 

  4. 4.

    Berstein, S.: Demonstration du theorems de Weierstrass, fonde sur le probabilities. Commun. Soc. Math. Kharkow. 13(1–2), (1912–1913)

  5. 5.

    Bleimann, G., Butzer, P.L., Hann, L.: A Bernstein-type operator approximating continuous functions on the semi-axis. Indag. Math. 42, 255–262 (1980)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cai, Q.B., Zeng, X.M.: On the convergence of a kind of q-gamma operators. J. Inequal. Appl. 2013, 105 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cai, Q.B., Zeng, X.M.: On the convergence of a kind of a modified q-gamma operators. J. Comput. Anal. Appl. 15(5), 826–863 (2013)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Cai, Q.B., Zhou, G.R.: On \((p,q)\)-analogue of Kantorovich type Bernstein–Stancu–Schurer operators. Appl. Math. Comput. 276, 12–20 (2016)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Chen, S.N., Cheng, W.T., Zeng, X.M.: Stancu type generalization of modified Gamma operators based on q-integers. Bull. Korean Math. Soc. 54(2), 359–373 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Devore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)

    Google Scholar 

  11. 11.

    Dogru, O., Gupta, V.: Monotonicity and the asymptotic estimate of Bleimann Butzer and Hahn operators on q-integers. Georgian Math. J. 12, 415–422 (2005)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Dogru, O., Gupta, V.: Korovkin-type approximation properties of bivariate q-Meyer–König and Zeller operators. Calcolo 42, 51–63 (2006)

    Article  Google Scholar 

  13. 13.

    Finta, Z., Gupta, V.: Approximation properties of q-Baskakov operators. Cent. Eur. J. Math. 8(1), 199–211 (2009)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gadzhiev, A.D.: Theorem of the type of P. P. Korovkin type theorems. Mat. Zametki 20(5), 781–786 (1976)

    MathSciNet  Google Scholar 

  15. 15.

    Govil, N.K., Gupta, V.: Convergence of q-Meyer–König–Zeller–Durrmeyer operators. Adv. Stud. Contemp. Math. 19, 97–108 (2009)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Gupta, V.: \((p,q)\)-Szász–Mirakjan–Baskakov operators. Complex Anal. Oper. Theory 12(1), 17–25 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Gupta, V., Agarwal, R.P.: Convergence Estimates in Approximation Theory. Springer, New York (2014)

    Google Scholar 

  18. 18.

    Gupta, V., Aral, A.: Convergence of the q analogue of Szász-beta operators. Appl. Math. Comput. 216, 374–380 (2010)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Gupta, V., Aral, A.: Some approximation properties of q-Baskakov Durrmeyer operators. Appl. Math. Comput. 218(3), 783–788 (2011)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Gupta, V., Radu, C.: Statistical approximation properties of q-Baskakov Kantorovich operators. Cent. Eur. J. Math. 7(4), 809–818 (2009)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Gupta, V., Rassias, T.M., Agrawal, P.N., Acu, A.M.: Recent Advances in Constructive Approximation Theory. Springer, New York (2018)

    Google Scholar 

  22. 22.

    Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)

    Google Scholar 

  23. 23.

    Karsli, H.: Rate of convergence of a new Gamma type operators for the functions with derivatives of bounded variation. Math. Comput. Model. 45(5–6), 617–624 (2007)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Karsli, H., Agrawal, P.N., Goyal, M.: General Gamma type operators based on q-integers. Appl. Math. Comput. 251, 564–575 (2015)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Karsli, H., Ali, Ö.M.: Direct local and global approximation results for the operators of gamma type. Hacet. J. Math. Stat. 39(2), 241–253 (2010)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Korovkin, P.P.: On convergence of linear operators in the space of continuous functions (Russian). Dokl. Akad. Nauk SSSR 90, 961–964 (1953)

    Google Scholar 

  27. 27.

    Lupas, A.: A q-analogue of the Bernstein operator. In: Seminar on Numerical and Statistical Calculus. University of Cluj-Napoca, vol. 9, pp. 85–92 (1987)

    Google Scholar 

  28. 28.

    Mahmoodov, N.I.: On q-parametric Szász–Mirakjan operators. Mediterr. J. Math. 7(3), 297–311 (2010)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Mahmoodov, N.I., Gupta, V., Kaffaoglu, H.: On certain q-Phillips operators. Rocky Mt. J. Math. 42(4), 1291–1312 (2012)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Malik, N., Gupta, V.: Approximation by \((p,q)\)-Baskakov-Beta operators. Appl. Math. Comput. 293, 49–53 (2017)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Meyer-König, W., Zeller, K.: Bernsteinsche Potenzreihen. Stud. Math. 19, 89–94 (1960)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Mirakjan, G.M.: Approximation des fonctions continues au moyen de polynomes de la forme \(e^{-nx}\sum k=0^{m_{n}}C_{k,n}x^{k}\) [Approximation of continuous functions with the aid of polynomials of the form \(e^{-nx}\sum k=0^{m_{n}}C_{k,n}x^{k}\)] (in French). Comptes rendus de l’Acad. des Sci. del’URSS 31, 201–205 (1941)

    Google Scholar 

  33. 33.

    Mursaleen, M., Khan, F., Khan, A.: On \((p,q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266, 874–882 (2015) (Erratum: Appl. Math. Comput. 278, 70–71 (2016))

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Mursaleen, M., Khan, F., Khan, A.: Some approximation results by \((p,q)\)-analogue of Bernstein–Stancu operators. Appl. Math. Comput. 264, 392–402 (2015) (Erratum: Appl. Math. Comput. 269, 744–746 (2016))

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Mursaleen, M., Nasiruzzaman, M., Nurgali, A.: Some approximation results on Bernstein–Schurer operators defined by \((p,q)\)-integers. J. Inequal. Appl. 2015, 249 (2015)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Phillips, G.M.: Bernstein polynomials based on q-integers. Ann. Numer. Math. 4, 511–518 (1997)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Szász, O.: Generalization of S. Bernstein’s polynomial to the infinite interval. J. Res. Natl. Bur. Stand. 45, 239–245 (1950)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Yuksel, I., Ispir, N.: Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl. 52(10–11), 1463–1470 (2006)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Zhao, C., Cheng, W.T., Zeng, X.M.: Some approximation properties of a kind of q-gamma Stancu. J. Inequal. Appl. 2014, 94 (2014)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the editor and anonymous referees for their helpful comments and suggestions.

Availability of data and materials

No data were used to support this study.

Funding

This research is supported by the National Natural Science Foundation of China (Grant No. 11626031 and Grant No. 11601266), the Philosophy and Social Sciences General Planning Project of Anhui Province of China (Grant No. AHSKYG2017D153), the Natural Science Foundation of Anhui Province of China (Grant No. 1908085QA29), the Natural Science Foundation of Fujian Province of China (Grant No. 2016J05017), the Natural Science Foundation of Anhui Province of China (Grant No. 1908085QA29), the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant No. 2018C087R), and the Program for New Century Excellent Talents in Fujian Province University. We also thank Fujian Provincial Key Laboratory of Data-Intensive Computing, Fujian University Laboratory of Intelligent Computing and Information Processing and Fujian Provincial Big Data Research Institute of Intelligent Manufacturing of China.

Author information

Affiliations

Authors

Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Wen-Tao Cheng.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cheng, W., Zhang, W. & Cai, Q. \((p,q)\)-gamma operators which preserve \(x^{2}\). J Inequal Appl 2019, 108 (2019). https://doi.org/10.1186/s13660-019-2053-3

Download citation

MSC

  • 41A10
  • 41A25
  • 41A36

Keywords

  • \((p,q)\)-integers
  • \((p,q)\)-gamma operators
  • Modulus of continuity
  • Rate of convergence