1 Introduction

The q-calculus has attracted attention of many researchers because of its applications in various fields such as numerical analysis, computer-aided geometric design, differential equations, and so on. In the field of approximation theory, the application of q-calculus has been the area of many recent researches.

Lupaş [1] presented the first q-analogue of the classical Bernstein operators in 1987. He studied the approximation and shape-preserving properties of these operators. Another q-companion of the classical Bernstein polynomials is due to Phillips [2]. Inspired by this, several authors produced generalizations of well-known positive linear operators based on q-integers and studied them extensively. For instance, the approximation properties of the Kantorovich-type q-Bernstein operators [3], q-BBH operators [4], q-analogue of generalized Bernstein-Schurer operators [5], weighted statistical approximation by Kantorovich-type q-Szász-Mirakjan operators [6], q-Szász-Durrmeyer operators [7], operators constructed by means of q-Lagrange polynomials and A-statistical approximation [8], statistical approximation properties of modified q-Stancu-Beta operators [9], and q-Bernstein-Schurer-Kantorovich operators [10].

The q-calculus has led to the discovery of the \((p,q)\)-calculus. Recently, Mursaleen et al. have used the \((p,q)\)-calculus in approximation theory. They have applied it to construct a \((p,q)\)-analogue of the classical Bernstein operators [11], a \((p,q)\)-analogue of the Bernstein-Stancu operators [12], and a \((p,q)\)-analogue of the Bernstein-Schurer operators [13] and have studied their approximation properties. Most recently, \((p,q)\)-analogues of some other operators have been studied in [14–18], and [19].

We now give some basic notions of the \((p,q)\)-calculus.

The \((p,q)\)-integer is defined by

$$ [n]_{p,q}=\frac{p^{n}-q^{n}}{p-q}, \quad n=0,1,2,\ldots, 0< q< p\leq1. $$

The \((p,q)\)-companion of the binomial expansion is

$$\begin{aligned}& (ax+by)_{p,q}^{n}=\sum_{k=0}^{n} \binom{n}{k}_{p,q}q^{\frac {k(k-1)}{2}}p^{\frac{(n-k)(n-k-1)}{2}}a^{n-k}b^{k}x^{n-k}y^{k},\\& (x+y)_{p,q}^{n}=(x+y) (px+qy) \bigl(p^{2}x+q^{2}y \bigr)\cdots\bigl(p^{n-1}x+q^{n-1}y\bigr). \end{aligned}$$

The \((p,q)\)-analogues of the binomial coefficients are defined by

$$ \binom{n}{k}_{p,q}=\frac{[n]_{p,q}!}{[k]_{p,q}![n-k]_{p,q}!}. $$

The \((p,q)\)-analogues of definite integrals of a function f are defined by

$$ \int_{0}^{a}f(x)\,d_{p,q}x=(q-p)a\sum _{k=0}^{\infty}\frac{p^{k}}{q^{k+1}} f \biggl( \frac{p^{k}}{q^{k+1}}a \biggr) \quad\mbox{when } \biggl\vert \frac {p}{q}\biggr\vert < 1 $$

and

$$ \int_{0}^{a}f(x)\,d_{p,q}x=(p-q)a\sum _{k=0}^{\infty}\frac{q^{k}}{p^{k+1}} f \biggl( \frac{q^{k}}{p^{k+1}}a \biggr) \quad \mbox{when } \biggl\vert \frac {q}{p}\biggr\vert < 1. $$

For \(m,n\in N\), the \((p,q)\)-gamma and the \((p,q)\)-beta functions are defined by

$$ \Gamma_{p,q}(n)= \int_{0}^{\infty}p^{\frac{n(n-1)}{2}}E_{p,q}(-qx)\,d_{p,q}x, \qquad \Gamma_{p,q}(n+1)=[n]_{p,q}! $$

and

$$ B_{p,q}(m,n)= \int_{0}^{\infty}\frac{x^{m-1}}{(1+x)^{m+n}}\,d_{p,q}x, $$
(1.1)

respectively. These two are related by

$$ B_{p,q}(m,n)=q^{\frac{2-m(m-1)}{2}}p^{\frac{-m(m-1)}{2}} \frac{\Gamma _{p,q}(n)\Gamma_{p,q}(m)}{\Gamma_{p,q}(m+n)}. $$
(1.2)

For \(p=1\), all the concepts of the \((p,q)\)-calculus reduce to those of q-calculus. The details on \((p,q)\)-calculus can be found in [20–22].

Stancu [23] introduced the beta operators to approximate the Lebesgue-integrable functions on \([0,\infty)\) as follows:

$$ L_{n}(f,x)=\frac{1}{B(nx,n+1)} \int_{0}^{\infty}\frac {t^{nx}}{(1+t)^{nx+n+1}}f(t)\,dt. $$

The q-companion of the Stancu-Beta operators was given by Aral and Gupta [24] as follows:

$$ L_{n}(f,x)=\frac{K(A,[n]_{q}x)}{B([n]_{q}x,[n]_{q}+1)} \int_{0}^{\infty /A}\frac{u^{[n]_{q}x-1}}{(1+u)^{[n]_{q}x+[n]_{q}+1}}f \bigl(q^{[n]_{q}x}u\bigr)\,d_{q}u. $$

Let \(0< q< p<1\). Mursaleen et al. [25] constructed the \((p,q)\)-Stancu-Beta operators as follows:

$$ L_{n}^{p,q}(f,x)=\frac{1}{B_{p,q}([n]_{p,q}x,[n]_{p,q}+1)} \int _{0}^{\infty}\frac{u^{[n]_{p,q}x-1}}{(1+u)^{[n]_{p,q}x+[n]_{p,q}+1}}f \bigl(p^{[n]_{p,q}x}q^{[n]_{p,q}x}u\bigr)\,d_{p,q}u. $$
(1.3)

They investigated the approximating properties and estimated the rate of convergence of these operators. Motivated by this work, we introduce the following sequence of operators:

$$\begin{aligned} S_{n,p,q}^{\alpha,\beta}(f;x)={}&\frac {1}{B_{p,q}([n]_{p,q}x,[n]_{p,q}+1)} \\ &{}\times\int_{0}^{\infty}\frac {u^{[n]_{p,q}x-1}}{(1+u)^{[n]_{p,q}x+[n]_{p,q}+1}}f \biggl( \frac{[n]_{p,q}p^{[n]_{p,q}x}q^{[n]_{p,q}x}u+\alpha}{[n]_{p,q}+\beta} \biggr) \,d_{p,q}u, \end{aligned}$$
(1.4)

where \(0\leq\alpha\leq\beta\). We call them two-parametric \((p,q)\)-Stancu-Beta operators. For \(\alpha=0=\beta\), the operators (1.4) coincide with the operators (1.3). So the latter is a generalization of the former.

2 Main results

We shall investigate approximation results for the operators (1.4). We calculate the moments of the operators \(S_{n,p,q}^{\alpha,\beta }(f;x)\) in the following lemma.

Lemma 2.1

Let \(S_{n,p,q}^{\alpha,\beta}(f;x)\) be given by (1.4). Then we have the following equalities:

  1. (i)

    \(S_{n,p,q}^{\alpha,\beta}(1;x)=1\),

  2. (ii)

    \(S_{n,p,q}^{\alpha,\beta}(t;x)=\frac{[n]_{p,q}}{([n]_{p,q}+\beta)}x+\frac{\alpha}{([n]_{p,q}+\beta)}\),

  3. (iii)

    \(S_{n,p,q}^{\alpha,\beta}(t^{2};x)=\frac{[n]_{p,q}^{3}}{pq([n]_{p,q}-1)([n]_{p,q}+\beta)^{2}}x^{2}+\frac{[n]_{p,q}}{([n]_{p,q}+\beta)^{2}} (\frac{[n]_{p,q}}{pq([n]_{p,q}-1)}+2\alpha )x+\frac{\alpha^{2}}{([n]_{p,q}+\beta)^{2}}\).

Proof

Using (1.1), (i) is immediate. Further,

$$\begin{aligned} S_{n,p,q}^{\alpha,\beta}(t;x) =&\frac {1}{B_{p,q}([n]_{p,q}x,[n]_{p,q}+1)}\\ &{}\times \int_{0}^{\infty}\frac {u^{[n]_{p,q}x-1}}{(1+u)^{[n]_{p,q}x+[n]_{p,q}+1}} \biggl( \frac{[n]_{p,q}p^{[n]_{p,q}x}q^{[n]_{p,q}x}u+\alpha}{([n]_{p,q}+\beta)} \biggr) \,d_{p,q}u \\ =&\frac{[n]_{p,q}}{([n]_{p,q}+\beta)}\frac {p^{[n]_{p,q}x}q^{[n]_{p,q}x}}{B_{p,q}([n]_{p,q}x,[n]_{p,q}+1)} \int_{0}^{\infty}\frac{u^{[n]_{p,q}x}}{ (1+u)^{[n]_{p,q}x+[n]_{p,q}+1}}\,d_{p,q}u \\ &{}+\frac{\alpha}{([n]_{p,q}+\beta)}\frac {1}{B_{p,q}([n]_{p,q}x,[n]_{p,q}+1)} \int_{0}^{\infty}\frac {u^{[n]_{p,q}x-1}}{(1+u)^{[n]_{p,q}x+[n]_{p,q}+1}}\,d_{p,q}u \\ =&\frac{[n]_{p,q}}{([n]_{p,q}+\beta)}L_{n}^{p,q}(t;x)+\frac{\alpha}{([n]_{p,q}+\beta)}L_{n}^{p,q}(1;x) \\ =&\frac{[n]_{p,q}}{([n]_{p,q}+\beta)}x+\frac{\alpha }{([n]_{p,q}+\beta)}, \end{aligned}$$

and (ii) is proved;

$$\begin{aligned} S_{n,p,q}^{\alpha,\beta}\bigl(t^{2};x\bigr) =& \frac{1}{B_{p,q}([n]_{p,q}x,[n]_{p,q}+1)}\\ &{}\times \int_{0}^{\infty}\frac {u^{[n]_{p,q}x-1}}{(1+u)^{[n]_{p,q}x+[n]_{p,q}+1}} \biggl( \frac{[n]_{p,q}p^{[n]_{p,q}x}q^{[n]_{p,q}x}u+\alpha}{([n]_{p,q}+\beta )} \biggr) ^{2}\,d_{p,q}u \\ =&\frac{[n]_{p,q}^{2}}{([n]_{p,q}+\beta)^{2}}\frac{p^{2[n]_{p,q}x}q^{2[n]_{p,q}x}}{B_{p,q}([n]_{p,q}x,[n]_{p,q}+1)}\int_{0}^{\infty}\frac {u^{[n]_{p,q}x+1}}{(1+u)^{[n]_{p,q}x+[n]_{p,q}+1}}\,d_{p,q}u \\ &{}+\frac{2\alpha}{[n]_{p,q}}{\bigl([n]_{p,q}+\beta\bigr)^{2}} \frac {q^{[n]_{p,q}x}}{B_{p,q}([n]_{p,q}x,[n]_{p,q}+1)} \int_{0}^{\infty}\frac{u^{[n]_{p,q}x}}{ (1+u)^{[n]_{p,q}x+[n]_{p,q}+1}}\,d_{p,q}u\\ &{}+ \frac{\alpha ^{2}}{([n]_{p,q}+\beta )^{2}} \frac{1}{B_{p,q}([n]_{p,q}x,[n]_{p,q}+1)} \int_{0}^{\infty }\frac{u^{[n]_{p,q}x-1}}{(1+u)^{[n]_{p,q}x+[n]_{p,q}+1}}\,d_{p,q}u \\ =&\frac{[n]_{p,q}^{2}}{([n]_{p,q}+\beta )^{2}}L_{n}^{p,q}\bigl(t^{2};x \bigr)+\frac{2\alpha[ n]_{p,q}}{([n]_{p,q}+\beta)^{2}}L_{n}^{p,q}(t;x)+\frac{ \alpha^{2}}{([n]_{p,q}+\beta)^{2}}L_{n}^{p,q}(1;x) \\ =&\frac{[n]_{p,q}^{2}}{([n]_{p,q}+\beta)^{2}} \biggl(\frac{[n]_{p,q}}{pq([n]_{p,q}-1)}x^{2}+ \frac{1}{pq([n]_{p,q}-1)}x \biggr)\\ &{}+\frac{2\alpha [ n]_{p,q}}{([n]_{p,q}+\beta)^{2}}+\frac{\alpha^{2}}{([n]_{p,q}+\beta)^{2}} \\ =&\frac{[n]_{p,q}^{3}}{pq([n]_{p,q}-1)([n]_{p,q}+\beta )^{2}}x^{2}+\frac{n}{([n]_{p,q}+\beta)^{2}} \biggl( \frac{[n]_{p,q}}{pq([n]_{p,q}-1)}+2\alpha \biggr)x\\ &{}+\frac{\alpha^{2}}{([n]_{p,q}+\beta)^{2}}, \end{aligned}$$

which proves (iii).

Hence, the lemma is proved. □

We readily obtain the following lemma.

Lemma 2.2

Let \(p,q\in(0,1)\). Then, for \(x\in[0,\infty)\), we have:

  1. (i)

    \(S_{n,p,q}^{\alpha,\beta}((t-x);x)=\frac{\alpha-\beta x}{([n]_{p,q}+\beta)}\),

  2. (ii)

    \(S_{n,p,q}^{\alpha,\beta}((t-x)^{2};x)\leq (\frac {[n]_{p,q}}{pq([n]_{p,q}-1)}-\frac{([n]_{p,q}-\beta)}{([n]_{p,q}+\beta)} )x^{2}+\frac{1}{pq([n]_{p,q}-1)}x+\frac{\alpha^{2}}{([n]_{p,q}+\beta )^{2}}\leq \frac{2(1+\beta)^{2}x^{2}+x+\alpha^{2}}{pq([n]_{p,q}-1)}\).

Proof

We have

$$\begin{aligned} S_{n,p,q}^{\alpha,\beta}\bigl((t-x);x\bigr) =&S_{n,p,q}^{\alpha,\beta }(t;x)-xS_{n,p,q}^{\alpha,\beta}(1;x) \\ =&\frac{[n]_{p,q}}{([n]_{p,q}+\beta)}x+\frac{\alpha }{([n]_{p,q}+\beta)}-x \\ =& \biggl(\frac{[n]_{p,q}}{([n]_{p,q}+\beta)}-1 \biggr)x+\frac{\alpha}{([n]_{p,q}+\beta)} \\ =&\frac{-\beta}{([n]_{p,q}+\beta)}x+\frac{\alpha}{([n]_{p,q}+\beta )} \\ =&\frac{\alpha-\beta x}{([n]_{p,q}+\beta)}, \end{aligned}$$

which proves (i). Now

$$\begin{aligned} &S_{n,p,q}^{\alpha,\beta}\bigl((t-x)^{2};x\bigr)\\ &\quad=S_{n,p,q}^{\alpha,\beta }\bigl(t^{2};x\bigr)+x^{2}S_{n,p,q}^{\alpha,\beta}(1;x)-2xS_{n,p,q}^{\alpha ,\beta }(t;x)\\ &\quad=\frac{[n]_{p,q}^{3}}{pq([n]_{p,q}-1)([n]_{p,q}+\beta )^{2}}x^{2}+\frac{[n]_{p,q}}{([n]_{p,q}+\beta)^{2}} \biggl( \frac{[n]_{p,q}}{pq([n]_{p,q}-1)} +2\alpha \biggr)x\\ &\qquad{}+\frac{\alpha^{2}}{([n]_{p,q}+\beta)^{2}}-2x \biggl(\frac{[n]_{p,q}}{([n]_{p,q}+\beta)}x+\frac{\alpha}{([n]_{p,q}+\beta)} \biggr)+x^{2}\\ &\quad=\frac{[n]_{p,q}^{3}}{pq([n]_{p,q}-1)([n]_{p,q}+\beta)^{2}}-\frac{2[n]_{p,q}}{([n]_{p,q}+\beta)+1}x^{2}+\frac{[n]_{p,q}^{2}}{pq([n]_{p,q}-1)([n]_{p,q}+\beta)^{2}}\\ &\qquad{}+\frac{2\alpha[ n]_{p,q}}{([n]_{p,q}+\beta)^{2}}-\frac {2\alpha}{([n]_{p,q}+\beta)}x+\frac{\alpha^{2}}{([n]_{p,q}+\beta)^{2}}\\ &\quad\leq \biggl(\frac{[n]_{p,q}}{pq([n]_{p,q}-1)}-\frac{([n]_{p,q}-\beta)}{ ([n]_{p,q}+\beta)} \biggr)x^{2}+ \frac{1}{pq([n]_{p,q}-1)}x+\frac{\alpha ^{2}}{([n]_{p,q}+\beta)^{2}}\\ &\quad=\frac{\{(p-q)[n]_{p,q}^{3}+([n]_{p,q}+pq[n]_{p,q}-pq)\beta ^{2}+(2\beta+pq)[n]_{p,q}^{2}\}x^{2}+([n]_{p,q}+\beta )^{2})x+pq([n]_{p,q}-1)\alpha^{2}}{pq([n]_{p,q}-1)([n]_{p,q}+\beta)^{2}} \\ &\quad=\frac{\{(p^{n}-q^{n})[n]_{p,q}^{2}+([n]_{p,q}+pq[n]_{p,q}-pq)\beta ^{2}+(2\beta+pq)[n]_{p,q}^{2}\}x^{2}+([n]_{p,q}+\beta )^{2})x+pq([n]_{p,q}-1)\alpha^{2}}{pq([n]_{p,q}-1)([n]_{p,q}+\beta)^{2}}\\ &\quad\leq\frac{2(\beta^{2}+\beta+1)x^{2}+x+\alpha^{2}}{pq([n]_{p,q}-1)}\\ &\quad\leq\frac{2(\beta+1)^{2}x^{2}+x+\alpha ^{2}}{pq([n]_{p,q}-1)} \end{aligned}$$

which gives (ii). Hence, the lemma is proved. □

Next, we present a direct theorem for the operators \(S_{n,p,q}^{\alpha ,\beta}(f;x)\).

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

$$ \| f\|=\sup_{0\leq x< \infty}\bigl|f(x)\bigr|. $$

Let \(\delta>0\) and \(W^{2}= \{ h:h^{\prime},h^{\prime\prime}\in C(I), I= [0, \infty) \} \), then the Peetre K-functional is defined by

$$ K_{2}(f,\delta)=\inf_{h\in W^{2}}\bigl\{ \| f-h\|+\delta\bigl\| h^{\prime\prime}\bigr\| \bigr\} . $$

The second-order modulus of continuity \(\omega_{2}\) of f is defined as

$$ \omega_{2}(f,\sqrt{\delta})=\sup_{0< p< \delta^{\frac{1}{2}}}\sup _{x\in I}\bigl|f(x+2p)-2f(x+p)+f(x)\bigr|. $$

By DeVore-Lorentz theorem (see [26], p.177, Theorem 2.4) there exists a constant \(C>0\) such that

$$ K_{2}(f,\delta)\leq C\omega_{2}(f,\sqrt{ \delta}). $$
(2.1)

Also, by \(\omega(f,\delta)\) we denote the first-order modulus of continuity of \(f\in C(I)\) defined as

$$ \omega(f,\delta)=\sup_{0< p< \delta}\sup_{x\in I}\bigl|f(x+p)-f(x)\bigr|. $$

We shall use the notation \(v^{2}(x)=x+x^{2}\).

Theorem 2.3

Suppose that \(f\in C_{B}[0,\infty)\) and \(0< p,q<1\). Then for all \(x\in[0,\infty)\) and \(n\geq2\), there exists a constant C such that

$$ \bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr|\leq C\omega_{2} \biggl(f, \frac {\delta _{n}(x)}{\sqrt{pq([n]_{p,q}-1)}} \biggr)+\omega \biggl(f,\frac{\gamma_{n}(x)}{[n]_{p,q}+\beta} \biggr), $$

where

$$ \delta_{n}^{2}(x)=v^{2}(x)+\frac{2pq\alpha^{2}}{([n]_{p,q}+\beta)} $$

and

$$ \gamma_{n(x)}^{2}= (\alpha-\beta x)^{2}+[n]_{p,q} \bigl([n]_{p,q}+\beta\bigr)x^{2} + \alpha \beta x. $$

Proof

Let us define the auxiliary operators

$$ S_{n,p,q}^{\ast\alpha,\beta}(f;x)=S_{n,p,q}^{\alpha,\beta }(f;x)-f \biggl(\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta} \biggr)+f(x). $$
(2.2)

By the Lemma 2.1 it is readily seen that these operators are linear and

$$ S_{n,p,q}^{\ast\alpha,\beta}\bigl((t-x);x\bigr)=0. $$
(2.3)

Suppose that \(g\in W^{2}\). By the Taylor expansion we can write

$$ g(t)=g(x)+g^{\prime}(x) (t-x)+ \int_{x}^{t}(t-u)g^{\prime\prime }(u)\,du,\quad t\in[0, \infty). $$

Operating by \(S_{n,p,q}^{\ast\alpha,\beta}(.;x)\) on both sides of the above and using (2.3), we obtain:

$$\begin{aligned}& S_{n,p,q}^{\ast\alpha,\beta}(g;x)=g(x)+S_{n,p,q}^{\ast\alpha,\beta } \biggl( \int_{x}^{t}(t-u)g^{\prime\prime}(u)\,du;x \biggr), \\& S_{n,p,q}^{\ast\alpha,\beta}(g;x)-g(x)=S_{n,p,q}^{\ast\alpha,\beta } \biggl( \int_{x}^{t}(t-u)g^{\prime\prime}(u)\,du;x \biggr), \\& \bigl|S_{n,p,q}^{\ast\alpha,\beta}(g;x)-g(x)\bigr|=\biggl\vert S_{n,p,q}^{\ast\alpha ,\beta} \biggl( \int_{x}^{t}(t-u)g^{\prime\prime}(u)\,du;x \biggr) \biggr\vert . \end{aligned}$$

Using (2.2) in the right-hand side, we get

$$\begin{aligned}[b] \bigl|S_{n,p,q}^{\ast\alpha,\beta}(g;x)-g(x)\bigr|={}& \biggl\vert S_{n,p,q}^{\alpha ,\beta } \biggl( \int_{x}^{t}(t-u)g^{\prime\prime}(u)\,du;x \biggr)\\ &{}- \int _{x}^{\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta}} \biggl(\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta}-u \biggr)g^{\prime\prime}(u)\,du \biggr\vert . \end{aligned} $$

So we obtain

$$\begin{aligned} &\bigl|S_{n,p,q}^{\ast\alpha,\beta}(g;x)-g(x)\bigr| \\ &\quad\leq \biggl|S_{n,p,q}^{\alpha,\beta} \biggl( \int_{x}^{t}(t-u)g^{\prime\prime }(u)\,du;x \biggr) \biggr|+ \biggl| \int_{x}^{\frac{[n]_{p,q}x+\alpha }{[n]_{p,q}+\beta}} \biggl(\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta}-u \biggr)g^{\prime\prime}(u)\,du \biggr| \\ &\quad\leq S_{n,p,q}^{\alpha,\beta} \biggl( \biggl| \int _{x}^{t}(t-u)g^{\prime \prime}(u)\,du \biggr|;x \biggr) + \int_{x}^{\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta}} \biggl|\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta }-u \biggr|\bigl|g^{\prime\prime}(u)\bigr|\,du . \end{aligned}$$

Using the linearity of the integral operator and the operator \(S_{n,p,q}^{\alpha ,\beta}(\cdot;x)\) in the second and first parts of right-hand side, respectively, and using the fact that for all \(x \in[0, \infty)\),

$$ \bigl|g(x)\bigr|\leq\| g\|, $$

we obtain

$$\begin{aligned} \bigl|S_{n,p,q}^{\ast\alpha,\beta}(g;x)-g(x)\bigr| \leq& \bigl\| g^{\prime \prime }\bigr\| S_{n,p,q}^{\alpha,\beta}\bigl((t-x)^{2};x \bigr) + \bigl\| g^{\prime \prime}\bigr\| \int_{x}^{\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta}} \biggl|\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta}-u \biggr|\,du. \end{aligned}$$
(2.4)

In the first part, solving the integral \(\int_{x}^{t} | t-u|\,du\) and using the linearity of the operators \(S_{n,p,q}^{\alpha,\beta}(\cdot;x)\), we readily see that

$$ S_{n,p,q}^{\alpha,\beta} \biggl( \int_{x}^{t}| t-u |\,du \biggr) \leq S_{n,p,q}^{\alpha,\beta} \bigl((t-x)^{2}; x \bigr), $$

and after some calculations, for the second part of (2.4), we get

$$\begin{aligned} &\int_{x}^{\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta}} \biggl|\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta}-u \biggr|\,du \\ &\quad\leq \frac{([n]_{p,q}x+\alpha)^{2}-x([n]_{p,q}x+\alpha)([n]_{p,q}+\beta)+ x^{2}([n]_{p,q}+\beta)^{2}}{([n]_{p,q}+\beta)^{2}}\\ &\quad=\frac{(\alpha-\beta x)^{2}+[n]_{p,q}x^{2}([n]_{p,q}+\beta)+\alpha\beta x}{([n]_{p,q}+\beta)^{2}}\\ &\quad= \biggl(\frac{\alpha-\beta x}{[n]_{p,q}+\beta} \biggr)^{2}+\frac {[n]_{p,q}}{[n]_{p,q}+\beta}x^{2}+ \frac{\alpha\beta}{([n]_{p,q}+\beta)^{2}}x. \end{aligned}$$

So by (2.4), we obtain

$$\begin{aligned} &\bigl|S_{n,p,q}^{\ast\alpha,\beta}(g;x)-g(x)\bigr| \\ &\quad\leq \bigl\| g^{\prime \prime }\bigr\| \biggl(S_{n,p,q}^{\alpha,\beta}\bigl((t-x)^{2};x \bigr) + \biggl(\frac{\alpha-\beta x}{[n]_{p,q}+\beta} \biggr)^{2}+\frac{[n]_{p,q}}{[n]_{p,q}+\beta }x^{2}+ \frac{\alpha\beta}{([n]_{p,q}+\beta)^{2}}x \biggr). \end{aligned}$$
(2.5)

Using Lemma 2.2(ii), we obtain

$$\begin{aligned} &S_{n,p,q}^{\alpha,\beta}\bigl((t-x)^{2};x\bigr)+ \biggl( \frac{\alpha-\beta x}{[n]_{p,q}+\beta} \biggr)^{2}+ \frac{[n]_{p,q}}{([n]_{p,q}+\beta)}x^{2}+ \frac{ \alpha\beta}{([n]_{p,q}+\beta)^{2}}x \\ &\quad\leq \biggl(\frac{[n]_{p,q}}{pq([n]_{p,q}-1)}-\frac{([n]_{p,q}-\beta)}{ ([n]_{p,q}+\beta)} \biggr)x^{2}+ \frac{1}{pq([n]_{p,q}-1)}x+\frac{\alpha ^{2}}{([n]_{p,q}+\beta)^{2}} \\ &\qquad{}+ \biggl(\frac{\alpha-\beta x}{[n]_{p,q}+\beta} \biggr)^{2}+ \frac {[n]_{p,q}}{([n]_{p,q}+\beta)}x^{2}+\frac{\alpha\beta}{([n]_{p,q}+\beta)^{2}}x \\ &\quad\leq\frac{(p-q)[n]_{p,q}^{3}}{pq([n]_{p,q}+\beta )^{2}([n]_{p,q}-1)}x^{2}+\frac{[n]_{p,q}^{2}+4pq(1-[n]_{p,q})\alpha\beta}{pq([n]_{p,q}+\beta )^{2}([n]_{p,q}-1)}x+ \frac{2\alpha^{2}}{([n]_{p,q}+\beta)^{2}} \\ &\quad\leq\frac {(p-q)[n]_{p,q}^{3}x^{2}+[n]_{p,q}^{2}x+2pq([n]_{p,q}-1)\alpha ^{2}}{pq([n]_{p,q}+\beta)^{2}([n]_{p,q}-1)} \\ &\quad=\frac{(p^{n}-q^{n})[n]_{p,q}^{2}x^{2}+[n]_{p,q}^{2}x+2pq([n]_{p,q}-1) \alpha^{2}}{pq([n]_{p,q}+\beta)^{2}([n]_{p,q}-1)} \\ &\quad\leq\frac{[n]_{p,q}^{2}x^{2}+[n]_{p,q}^{2}x+2pq[n]_{p,q}\alpha^{2}}{ pq([n]_{p,q}+\beta)^{2}([n]_{p,q}-1)} \\ &\quad\leq\frac{[n]_{p,q}(1+x)x+2pq\alpha^{2}}{pq([n]_{p,q}+\beta )^{2}([n]_{p,q}-1)} \\ &\quad\leq\frac{1}{pq([n]_{p,q}-1)} \biggl(v^{2}(x)+\frac{2pq\alpha^{2}}{([n]_{p,q}+\beta)} \biggr) \\ &\quad=\frac{\delta_{n}^{2}(x)}{pq([n]_{p,q}-1)}, \end{aligned}$$

where

$$ \delta_{n}^{2}(x)= v^{2}(x)+\frac{2pq\alpha^{2}}{([n]_{p,q}+\beta)}. $$

Therefore, by (2.5) we get

$$ \bigl|S_{n,p,q}^{\ast\alpha,\beta}(g;x)-g(x)\bigr|\leq\frac{\delta _{n}^{2}(x)}{pq([n]_{p,q}-1)} \bigl\| g^{\prime\prime}\bigr\| . $$
(2.6)

On the other hand, by (2.2) we have

$$ \bigl|S_{n,p,q}^{\ast\alpha,\beta}(f;x)\bigr|\leq \bigl|S_{n,p,q}^{\alpha,\beta }(f;x)\bigr|+2 \| f\|\leq3\| f\|. $$
(2.7)

By (2.2), (2.6), and (2.7), we obtain:

$$\begin{aligned} \bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr| \leq&\bigl|S_{n,p,q}^{\ast\alpha ,\beta }(f-g;x)-(f-g) (x)\bigr|+\bigl|S_{n,p,q}^{\ast\alpha,\beta}(g;x)-g(x)\bigr| \\ &{}+\biggl|f \biggl(\frac{[n]_{p,q}x+\alpha}{[n]_{p,q}+\beta} \biggr)-f(x)\biggr| \\ \leq&4\| f-g\|+\frac{\delta_{n}^{2}(x)}{pq([n]_{p,q}-1)}\bigl\| g^{\prime\prime}\bigr\| \\ &{}+\omega \biggl(f,\frac{\sqrt{(\alpha-\beta x)^{2}+[n]_{p,q}([n]_{p,q}+\beta)x^{2}+\alpha\beta x}}{[n]_{p,q}+\beta } \biggr) \\ =& 4\| f-g\|+\frac{\delta_{n}^{2}(x)}{pq([n]_{p,q}-1)} \bigl\| g^{\prime\prime}\bigr\| +\omega \biggl(f, \frac{\gamma _{n}(x)}{[n]_{p,q}+\beta} \biggr), \end{aligned}$$
(2.8)

where

$$ \gamma_{n(x)}^{2} =(\alpha-\beta x)^{2}+[n]_{p,q} \bigl([n]_{p,q}+\beta \bigr)x^{2}+\alpha \beta x. $$

Taking the infimum over all \(g\in W^{2}\) on the right-hand side of (2.8), we obtain

$$ \bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr|\leq CK_{2} \biggl(f, \frac{\delta _{n}^{2}(x)}{pq([n]_{p,q}-1)} \biggr)+\omega \biggl(f,\frac{(\gamma_{n})x}{[n]_{p,q}+\beta} \biggr). $$

Using relation (2.1), for \(p,q\in(0,1)\), we get

$$ \bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr|\leq C\omega_{2} \biggl(f, \frac {\delta _{n}(x)}{\sqrt{pq([n]_{p,q}-1)}} \biggr)+\omega \biggl(f,\frac{\gamma_{n}(x)}{[n]_{p,q}+\beta} \biggr), $$

and this completes the proof. □

3 Rate of approximation

Let \(B_{x^{2}}[0,\infty)\) denote the set of all functions f such that \(f(x)\leq M_{f}(1+x^{2})\), where \(M_{f}\) is a constant depending on f. By \(C_{x^{2}}[0,\infty)\) we denote the subspace of all continuous functions in the space \(B_{x^{2}}[0,\infty)\). Also, we denote by \(C_{x^{2}}^{\ast }[0,\infty)\), 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 finite with

$$ \| f\|=\sup_{x\in[0,\infty)}\frac{|f(x)|}{1+x^{2}}. $$

For \(a>0\), the modulus of continuity of f over \([0,a]\) is defined by

$$ \omega_{a}(f,\delta)=\sup_{|t-x|\leq\delta} \sup _{0\leq x,t\leq a}\bigl|f(t)-f(x)\bigr|. $$

We have the following proposition.

Proposition 3.1

  1. (i)

    For \(f\in C_{x^{2}}[0,\infty)\), the modulus of continuity \(\omega _{a}(f,\delta)\), \(a>0\), approaches to zero.

  2. (ii)

    For every \(\delta>0\), we have

    $$ \bigl|f(y)-f(x)\bigr| \leq \biggl(1+\frac{|y-x|}{\delta} \biggr)\omega_{a}(f, \delta) $$

    and

    $$ \bigl|f(y)-f(x)\bigr| \leq \biggl(1+\frac{(y-x)^{2}}{\delta^{2}} \biggr)\omega _{a}(f, \delta). $$

In the following theorem, we estimate the rate of convergence of the operators \(S_{n,p,q}^{\alpha,\beta}(f;x)\).

Theorem 3.2

Let \(f\in C_{x^{2}}[0,\infty),p,q\in(0,1)\), and let \(\omega _{a+1}(f,\delta)\) be the modulus of continuity on the interval \([0,1+a]\subseteq[0,\infty),a>0\). Then, for \(n\geq2\), we have

$$\begin{aligned} \bigl\| S_{n,p,q}^{\alpha,\beta}(f)-f\bigr\| _{C[0,a]}\leq{}& \frac{4M_{f}(1+a^{2})(2(1+\beta)^{2}a^{2}+a+\alpha^{2})}{pq([n]_{p,q}-1)}\\ &{}+2\omega_{1+a} \biggl(f, \biggl( \frac{2(1+\beta)^{2}a^{2}+a+\alpha^{2}}{pq([n]_{p,q}-1)} \biggr)^{\frac{1}{2}} \biggr). \end{aligned}$$

Proof

Let \(x\in[0,a]\) and \(t> a+1\). Since \(1+x< t\), we have

$$\begin{aligned} \bigl|f(t)-f(x)\bigr|&\leq M_{f}\bigl(x^{2}+t^{2}+2 \bigr)\leq M_{f}\bigl(2+3x^{2}+2(t-x)^{2}\bigr) \\ &\leq M_{f}\bigl(4+3x^{2}\bigr) (t-x)^{2} \leq4M_{f}\bigl(1+a^{2}\bigr) (t-x)^{2}. \end{aligned}$$
(3.1)

For \(\delta>0\), \(x\in[0,a]\), \(t-1\leq a\), by Proposition 3.1 we obtain

$$ \bigl|f(t)-f(x)\bigr|\leq\omega_{1+a}\bigl(f,|t-x|\bigr)\leq \omega_{1+a}(f,\delta) \biggl(1+\frac{1}{\delta}|t-x| \biggr). $$
(3.2)

By (3.1) and (3.2), for \(x\in[0,a]\) and nonnegative t, we can write

$$ \bigl|f(t)-f(x)\bigr|\leq4M_{f}\bigl(1+a^{2}\bigr) (t-x)^{2}\omega_{1+a}(f,\delta) \biggl(1+\frac{1}{\delta}|t-x| \biggr). $$
(3.3)

Therefore,

$$\begin{aligned} &\bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr| \\ &\quad\leq S_{n,p,q}^{\alpha,\beta } \bigl(\bigl|f(t)-f(x)\bigr|;x\bigr) \\ &\quad\leq4M_{f}\bigl(1+a^{2}\bigr)S_{n,p,q}^{\alpha,\beta} \bigl((t-x)^{2};x\bigr)+\omega _{1+a}(f,\delta) \biggl(1+ \frac{1}{\delta}\bigl(S_{n,p,q}^{\alpha,\beta }\bigl((t-x)^{2};x \bigr)\bigr)^{\frac{1}{2}} \biggr). \end{aligned}$$

Hence, using the Lemma 2.2(ii) and the Schwarz inequality, for every \(p,q\in(0,1)\) and \(x\in[0,a]\), we obtain

$$\begin{aligned} \bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr| \leq&4M_{f} \bigl(1+a^{2}\bigr) \biggl(\frac{ 2(1+\beta)^{2}x^{2}+x+\alpha^{2}}{pq([n]_{p,q}-1)} \biggr) \\ &{}+\omega_{1+a}(f,\delta) \biggl(1+\frac{1}{\delta} \biggl( \frac{2(1+\beta )^{2}a^{2}+a+\alpha^{2}}{pq([n]_{p,q}-1)} \biggr)^{\frac{1}{2}} \biggr) \\ \leq&\frac{4M_{f}(1+a^{2})(2(1+\beta)^{2}a^{2}+a+\alpha^{2})}{pq([n]_{p,q}-1)} \\ &{}+\omega_{1+a} \biggl(1+\frac{1}{\delta} \biggl( \frac{2(1+\beta )^{2}a^{2}+a+\alpha^{2}}{pq([n]_{p,q}-1)} \biggr)^{\frac{1}{2}} \biggr). \end{aligned}$$

By choosing \(\delta^{2}=\frac{2(1+\beta)^{2}a^{2}+a+\alpha^{2}}{pq([n]_{p,q}-1)}\) we get the required result. □

4 Weighted approximation

This section is devoted to the study of weighted approximation theorems for the operators (2.2).

Theorem 4.1

Suppose that \(p=p_{n}\) and \(q=q_{n}\) are two sequences satisfying \(0< p_{n},q_{n}<1\) and suppose that \(p_{n}\rightarrow1\) and \(q_{n}\rightarrow1\) as \(n\rightarrow \infty\). Then, for each \(f\in C_{x^{2}}^{\ast}[0,\infty)\), we have

$$ \lim_{n\rightarrow\infty}\bigl\| S_{n,p_{n},q_{n}}^{\alpha,\beta }(f)-f \bigr\| _{x^{2}}=0. $$

Proof

By the theorem in [27] it suffices to prove that

$$ \lim_{n\rightarrow\infty}\bigl\| S_{n,p_{n},q_{n}}^{\alpha,\beta } \bigl(t^{i}\bigr)-x^{i}\bigr\| _{x^{2}}=0 \quad\mbox{for } i=0,1,2. $$
(4.1)

By Lemma 2.1(i)-(ii), the conditions of (4.1) are easily verified for \(i=0\) and 1. For \(i=2\), we can write

$$\begin{aligned} &\bigl\| S_{n,p_{n},q_{n}}^{\alpha,\beta}\bigl(t^{2}\bigr)-x^{2} \bigr\| _{x^{2}} \\ &\quad=\sup_{x\in[0,\infty)}\frac{|S_{n,p_{n},q_{n}}^{\alpha,\beta }(t^{2})-x^{2}|}{1+x^{2}} \\ &\quad\leq \biggl(\frac{[n]_{p_{n},q_{n}}^{3}}{p_{n}q_{n}([n]_{p_{n},q_{n}}-1)([n]_{p_{n},q_{n}}+\beta)^{2}}-1 \biggr)\sup_{x\in[0,\infty)} \frac{x^{2}}{1+x^{2}} \\ &\qquad{}+\frac{[n]_{p_{n},q_{n}}^{2}+2p_{n}q_{n}[n]_{p_{n},q_{n}}([n]_{p_{n},q_{n}}-1)\alpha}{p_{n}q_{n}([n]_{p_{n},q_{n}}-1)([n]_{p_{n},q_{n}}+\beta)^{2}}\sup_{x\in[0,\infty)} \frac{x}{1+x^{2}}+\frac{\alpha^{2}}{([n]_{p_{n},q_{n}}+\beta)^{2}} \\ &\quad\leq\frac {(p_{n}^{n}-q_{n}^{n})[n]_{p_{n},q_{n}}^{2}-p_{n}q_{n}(2\beta -1)[n]_{p_{n},q_{n}}^{2}-q_{n}\beta(\beta -1)[n]_{p_{n},q_{n}}+q_{n}\beta ^{2}}{p_{n}q_{n}([n]_{p_{n},q_{n}}-1)([n]_{p_{n},q_{n}}+\beta)^{2}} \\ &\qquad{}+ \biggl(\frac{[n]_{p_{n},q_{n}}^{2}+2p_{n}q_{n}[n]_{p_{n},q_{n}}([n]_{p_{n},q_{n}}-1)\alpha}{p_{n}q_{n}([n]_{p_{n},q_{n}}-1)([n]_{p_{n},q_{n}}+\beta )^{2}} \biggr)+\frac{\alpha^{2}}{([n]_{p_{n},q_{n}}+\beta)^{2}}, \end{aligned}$$

which implies that

$$ \lim_{n\rightarrow\infty}\bigl\| S_{n,p_{n},q_{n}}^{\alpha,\beta } \bigl(t^{2},x\bigr)-x^{2}\bigr\| _{x^{2}}=0. $$

This completes the proof of the theorem. □

Theorem 4.2

Let \(p=(p_{n})\) and \(q=(q_{n})\) be two sequences such that \(0< p_{n}, q_{n}<1\), and let \(p_{n}\rightarrow1\) and \(q_{n}\rightarrow1\) as \(n\rightarrow \infty\). Then, for each \(f\in C_{x^{2}}[0,\infty)\) and all \(\alpha>0\), we have

$$ \lim_{n\rightarrow\infty}\sup_{x\in[0,\infty)}\frac{|S_{n,p_{n},q_{n}}^{\alpha,\beta}(f;x)-f(x)|}{(1+x^{2})^{1+\alpha^{2}}}=0. $$

Proof

For \(x_{0}>0\) fixed, we have:

$$\begin{aligned} \sup_{x\in[0,\infty)}\frac{|S_{n,p_{n},q_{n}}^{\alpha,\beta }(f;x)-f(x)|}{(1+x^{2})^{1+\alpha^{2}}} =&\sup_{x\leq x_{0}} \frac{|S_{n,p_{n},q_{n}}^{\alpha,\beta}(f;x)-f(x)|}{(1+x^{2})^{1+\alpha ^{2}}}+\sup_{x\geq x_{0}}\frac{|S_{n,p_{n},q_{n}}^{\alpha,\beta }(f;x)-f(x)|}{(1+x^{2})^{1+\alpha^{2}}} \\ \leq&\bigl\| S_{n,p_{n},q_{n}}^{\alpha,\beta}(f)-f\bigr\| _{C[0,a]}+\| f \|_{x^{2}}\sup_{x\geq x_{0}}\frac{|S_{n,p_{n},q_{n}}^{\alpha,\beta }(1+t^{2};x)|}{(1+x^{2})^{1+\alpha^{2}}} \\ &{}+\sup_{x\geq x_{0}}\frac{|f(x)|}{(1+x^{2})^{1+\alpha^{2}}}. \end{aligned}$$

The first term of this inequality goes to zero by Theorem 3.2. Also, for any fixed \(x_{0}>0\), it is readily seen from Lemma 2.1 that

$$ \sup_{x\geq x_{0}}\frac{|S_{n,p_{n},q_{n}}^{\alpha,\beta }(1+t^{2};x)|}{(1+x^{2})^{1+\alpha^{2}}} $$

approaches zero as \(n\rightarrow\infty\). If we choose \(x_{0}>0\) large enough so that the last part of the last inequality is arbitrarily small, then our theorem is proved. □

5 Voronovskaya-type theorem

This section presents the Voronovskaya-type theorem for the operators \(S_{n,p,q}^{\alpha,\beta}(f;x)\). We need the following lemma.

Lemma 5.1

Suppose that \(p_{n},q_{n}\in(0,1)\) are such that \(p_{n}^{n}\rightarrow a,q_{n}^{n}\rightarrow b \) (\(0\leq a,b<1\)) as \(n\rightarrow\infty\). Then, for every \(x\in[0,\infty)\), simple computations yield

$$\begin{aligned}& \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}}^{\alpha ,\beta } \bigl((t-x);x\bigr)=\alpha-\beta x, \\& \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}}^{\alpha ,\beta } \bigl((t-x)^{2};x\bigr)=(1-a) (1-b)x^{2}+x. \end{aligned}$$

Theorem 5.2

Assume that \(p_{n},q_{n}\in(0,1)\) are such that \(p_{n}^{n}\rightarrow a,q_{n}^{n}\rightarrow b \) (\(0\leq a,b<1\)) as \(n\rightarrow\infty\). Then, for \(f\in C_{x^{2}}^{\ast}[0,\infty)\) such that \(f^{\prime },f_{x^{2}}^{\prime \prime\ast}[0,\infty)\), we have

$$ \lim_{n\rightarrow\infty}[n]_{p_{n},q_{n}}\bigl(S_{n,p_{n},q_{n}}^{\alpha ,\beta}(f;x)-f(x) \bigr)=(\alpha-\beta x)f^{\prime}(x)+\frac {(1-a)(1-b)x^{2}+x}{2}f^{\prime\prime}(x) $$

uniformly on \([0,A]\) for any \(A>0\).

Proof

Let \(f, f', f''\in C^{*}_{x^{2}}[0,\infty)\) and \(x \in[0, \infty)\). By the Taylor formula we can write

$$ f(t)= f(x)+ (t-x)f'(x)+\frac{1}{2}(t-x)^{2} f''(x)+ r(t; x) (t-x)^{2}, $$
(5.1)

where \(r(t;x)\) is the remainder term, \(r(\cdot;x) \in C^{*}_{x^{2}}[0,\infty)\), and \(\lim_{t\rightarrow x} r(t;x)=0\). Operating by \(S_{n,p_{n},q_{n}}^{\alpha,\beta}\) on both sides of (5.1), we get

$$\begin{aligned} &[n]_{p_{n},q_{n}}\bigl(S_{n,p_{n},q_{n}}^{\alpha,\beta}(f; x)-f(x)\bigr)\\ &\quad= [n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}}^{\alpha,\beta}\bigl((t-x);x \bigr)f'(x)+\frac{1}{2} [n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}}^{\alpha,\beta} \bigl((t-x)^{2};x\bigr)f''(x) \\ &\qquad{}+ [n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}}^{\alpha,\beta}\bigl(r(\cdot;x) ( \cdot-x)^{2};x\bigr). \end{aligned}$$

It follows from the Cauchy-Schwarz inequality that

$$ S_{n,p_{n},q_{n}}^{\alpha,\beta}\bigl(r(\cdot;x) ( \cdot-x)^{2};x\bigr)\leq\sqrt {S_{n,p_{n},q_{n}}^{\alpha,\beta}(r^{2}( \cdot;x);x}\sqrt{S_{n,p_{n},q_{n}}^{\alpha ,\beta}(r\bigl(( \cdot-x)^{4};x\bigr)}. $$
(5.2)

Note that \(r^{2}(x;x)=0\) and \(r^{2}(\cdot;x)\in C^{*}_{x^{2}}[0,\infty)\). Therefore, it follows that

$$ \lim_{n \rightarrow\infty}S_{n,p_{n},q_{n}}^{\alpha,\beta} \bigl(r^{2}(\cdot;x);x\bigr)= r^{2}(x;x)=0 $$
(5.3)

uniformly over \([0, A]\).

By Lemma 5.1 and equations (5.2) and (5.3), we obtain

$$ \lim_{n \rightarrow\infty}[n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}}^{\alpha,\beta } \bigl(r(\cdot;x) (\cdot-x)^{2};x\bigr)=0. $$

Thus, we obtain

$$\begin{aligned} &\lim_{n \rightarrow\infty}[n]_{p_{n},q_{n}}\bigl(S_{n,p_{n},q_{n}}^{\alpha,\beta}(f; x)-f(x)\bigr) \\ &\quad=\lim_{n \rightarrow\infty}\biggl([n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}}^{\alpha,\beta }\bigl((t-x);x\bigr)f'(x)+ \frac{1}{2}[n]_{p_{n},q_{n}} S_{n,p_{n},q_{n}}^{\alpha,\beta } \bigl((t-x)^{2};x\bigr)f''(x) \\ &\qquad{}+[n]_{p_{n},q_{n}}S_{n,p_{n},q_{n}}^{\alpha,\beta}\bigl(r(\cdot;x) ( \cdot-x)^{2};x\bigr)\biggr) \\ &\quad= (\alpha-\beta x)f'(x)+ \frac{(1-a)(1-b)x^{2}+x}{2} f''(x). \end{aligned}$$

 □

6 Pointwise estimates

In this section, we study pointwise estimates of rate of convergence of the operators \(S_{n,p,q}^{\alpha,\beta}(f;x)\).

Let \(0<\nu\leq\) and \(E\subset[0,\infty)\). We say that a function \(f\in C[0,\infty)\) belongs to \(Lip(\nu)\) if

$$ \bigl|f(t)-f(x)\bigr|\leq M_{f}|t-x|^{\nu}, \quad t \in[0,\infty), x\in E, $$
(6.1)

where \(M_{f}\) is a constant depending on α and f only.

We have the following theorem.

Theorem 6.1

Let \(\nu\in(0,1],f\in Lip(\nu)\), and \(E\subset[0,\infty)\). Then, for \(x\in[0,\infty)\),

$$\begin{aligned} &\bigl\| S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr\| \\ &\quad\leq M_{f} \biggl\{ \biggl( \biggl(\frac{[n]_{p,q}}{pq([n]_{p,q}-1)}-\frac{([n]_{p,q}-\beta)}{([n]_{p,q}+\beta)} \biggr)x^{2}+\frac{1}{pq([n]_{p,q}-1)}x+\frac{\alpha ^{2}}{([n]_{p,q}+\beta)^{2}} \biggr)^{\frac{\nu}{2}} \\ &\qquad{}+2\bigl(d(x,E)\bigr)^{\nu} \biggr\} , \end{aligned}$$

where \(d(x,E)\) denotes the distance of the point x from the set E, defined by

$$ d(x,E)=\inf\bigl\{ {|x-y|:y\in E}\bigr\} . $$

Proof

Taking \(y\in\bar{E}\), we can write

$$ \bigl|f(t)-f(x)\bigr|\leq\bigl|f(t)-f(y)\bigr|+\bigl|f(y)-f(x)\bigr|, \quad x\in[0,\infty). $$

By (6.1) we have

$$\begin{aligned}[b] \bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr| &=\bigl|S_{n,p,q}^{\alpha,\beta }(f;x)-S_{n,p,q}^{\alpha,\beta} \bigl(f(x);x\bigr)\bigr| \\ &\leq S_{n,p,q}^{\alpha,\beta}\bigl(\bigl|f(t)-f(x)\bigr|;x\bigr) \\ &\leq S_{n,p,q}^{\alpha,\beta}\bigl(\bigl|f(t)-f(y)\bigr|;x\bigr)+S_{n,p,q}^{\alpha ,\beta } \bigl(\bigl|f(y)-f(x)\bigr|;x\bigr) \\ &\leq S_{n,p,q}^{\alpha,\beta}\bigl(\bigl|f(t)-f(y)\bigr|;x\bigr)+\bigl|f(x)-f(y)\bigr| \\ &\leq M_{f}{S_{n,p,q}^{\alpha,\beta}\bigl(|t-y|^{\nu};x \bigr)+|x-y|^{\nu}} \\ &\leq M_{f}{S_{n,p,q}^{\alpha,\beta}\bigl(|t-x|^{\nu}+|x-y|^{\nu };x \bigr)+|x-y|^{\nu}} \\ &\leq M_{f}{S_{n,p,q}^{\alpha,\beta}\bigl(|t-x|^{\nu};x \bigr)+2|x-y|^{\nu}}. \end{aligned} $$

Using the Hölder inequality with \(p=\frac{2}{\nu},q=\frac{2}{2-\nu }\), we obtain

$$\begin{aligned} &\bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr| \\ &\quad\leq M_{f} \bigl\{ \bigl(S_{n,p,q}^{\alpha,\beta}\bigl(|t-x|^{p\nu};x\bigr) \bigr)^{\frac{1}{p}} \bigl(S_{n,p,q}^{\alpha,\beta} \bigl(1^{q};x\bigr) \bigr)^{\frac{1}{q}}+2\bigl(d(x,E) \bigr)^{\nu } )\bigr\} \\ &\quad=M_{f} \bigl\{ \bigl(S_{n,p,q}^{\alpha,\beta} \bigl(|t-x|^{2};x\bigr) \bigr)^{\frac{\nu}{2}}+2\bigl(d(x,E) \bigr)^{\nu} ) \bigr\} \\ &\quad= \biggl\{ \biggl( \biggl(\frac{[n]_{p,q}}{pq([n]_{p,q}-1)}-\frac{([n]_{p,q}-\beta)}{([n]_{p,q}+\beta)} \biggr)x^{2}+\frac {1}{pq([n]_{p,q}-1)}x+\frac{\alpha^{2}}{([n]_{p,q}+\beta)^{2}} \biggr)^{\frac{\nu}{2}} \\ &\qquad{}+2\bigl(d(x,E)\bigr)^{\nu} \biggr\} , \end{aligned}$$

and the theorem is proved. □

We now present a theorem regarding a local direct estimate for the operators \(S_{n,p,q}^{\alpha,\beta}(f;x)\) in terms of the Lipschitz-type maximal function of order ν as introduced by Lenze [28]. It is defined by

$$ \tilde{\omega}_{\nu}(f;x)=\sup_{y\neq x, y\in[0,\infty)} \frac{ |f(y)-f(x)|}{|y-x|^{\nu}},\quad x\in[0,\infty), \nu\in(0,1]. $$
(6.2)

Theorem 6.2

Let \(\nu\in(0,1]\) and \(f\in C[0,\infty)\). Then, for each \(x\in [ 0,\infty)\), we have

$$\begin{aligned} &\bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr|\\ &\quad\leq\tilde{\omega}_{\nu}(f;x) \biggl\{ \biggl(\frac{[n]_{p,q}}{pq([n]_{p,q}-1)}-\frac{([n]_{p,q}-\beta )}{([n]_{p,q}+\beta)} \biggr)x^{2}+ \frac{1}{pq([n]_{p,q}-1)}x+\frac{\alpha ^{2}}{([n]_{p,q}+\beta)^{2}} \biggr\} ^{\frac{\nu}{2}}. \end{aligned}$$

Proof

By (6.2) we can write

$$ \bigl|f(t)-f(x)\bigr|\leq\tilde{\omega}_{\nu}(f;x)|t-x|^{\nu} $$

and

$$ \bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr|\leq S_{n,p,q}^{\alpha,\beta } \bigl(\bigl|f(t)-f(x)\bigr|;x\bigr)\leq\tilde{\omega}_{\nu}(f;x)S_{n,p,q}^{\alpha,\beta } \bigl(|t-x|^{\nu};x\bigr). $$

Using the Lemma 2.2 and applying the Hölder inequality with \(p=\frac {2}{\nu}\), \(q= \frac{2}{2-\nu}\), we obtain

$$ \bigl|S_{n,p,q}^{\alpha,\beta}(f;x)-f(x)\bigr|\leq\tilde{\omega}_{\nu }(f;x)S_{n,p,q}^{\alpha,\beta} \bigl(|t-x|^{\nu};x\bigr), $$

which proves the theorem. □

Remark

The further properties of the operators such as convergence properties via summability methods (see, e.g., [29–31]) can be studied.

7 Conclusions

In this paper, we have introduced a two-parametric \((p,q)\)-analogue of the Stancu-Beta operators and studied some approximating properties of these operators. We also obtained the Voronovskaya-type estimate and the weighted approximation results for these operators. Furthermore, we obtained a pointwise estimate for these operators.