# Approximation by modified $$(p,q)$$-gamma-type operators

## Abstract

The main object of this paper is to construct a new class of modified $$(p,q)$$-Gamma-type operators. For this new class of operators, in section one, the general moments are find; in section two, the Korovkin-type theorem and some direct results are proved by considering the modulus of continuity and modulus of smoothness and their behavior in Lipschitz-type spaces. In section three, some results in the weighted spaces are given, and in the end, some shape-preserving properties are proven.

## 1 Introduction

One of the central theorems in the approximation theory is a Korovkin-type theorem. It is studied in various function spaces and in the various forms of convergence, starting from standard convergence [1, 12, 18, 27, 29], statistical convergence [3, 9, 10, 16, 23], power summability form of it [4â€“8, 24], and many other forms. In this paper, we will study the kind of the modified $$(p,q)$$-Gamma-type operators, and for these operators, we will prove the Korovkin-type theorem and some direct results by considering the modulus of continuity and modulus of smoothness and their behavior in Lipschitz-type spaces. In Sect.Â 3, some results in the weighted spaces are given, and in the end, some shape-preserving properties are proven. In [25], the following Gamma-type operators were introduced:

$$G_{n}(f,x)= \int _{0}^{\infty}K_{n}(x,u)f \biggl( \frac{n}{u} \biggr)\,du,$$
(1.1)

where

$$K_{n}(x,u)=\frac{x^{n+1}}{\Gamma (n+1)}e^{-xu}u^{n}, \quad x\in (0,\infty ).$$

Later one, in [29], the above operators have been modified to the following form:

$$\mathcal{G}_{n}(f,x)= \int _{0}^{\infty}K_{n}(x,u)f ( nu )\,du,$$
(1.2)

where

$$K_{n}(x,u)=\frac{x^{n+3}}{\Gamma (n+3)}e^{-\frac{x}{u}}u^{-n-4}, \quad x \in (0,\infty ).$$

Recently, in [21], the above operators have been modified as follows:

$$\mathcal{G}_{n,q}(f,x)= \int _{0}^{\frac{\infty}{A}}K_{n,q}(x,u)f \bigl([n]_{q}u\bigr) )\,d_{q}u,$$
(1.3)

where

$$\mathcal{K}_{n,q}(x,u)=\frac{qx^{n+1}}{\Gamma _{q}(n+1)}E(-qx/u)u^{-n-4}, \quad x\in (0,\infty ).$$

For any function f, the $$(p,q)$$-derivative is given by (for example, see [11, 19])

$$D_{p,q}f(x)=\frac{f(px)-f(qx)}{(p-q)x},\quad x\neq 0,$$

and in case where f is differentiable at 0, then $$D_{p,q}f(0)=f'(0)$$. We know that

$$[n]_{p,q}=\frac{p^{n}-q^{n}}{p-q},\qquad [n]_{p,q}!=\prod _{j=1}^{n}[j]_{p,q},\qquad [0]_{p,q}!=1, \qquad \binom{n}{k}_{p,q}= \frac{[n]_{p,q}!}{[k]_{p,q}![n-k]_{p,q}!},$$

for all $$0\leq k\leq n$$. In [13], it is proved that (Theorem 1)

$$\begin{bmatrix} {n+1} \\ {k}\end{bmatrix}_{pq} =p^{k} \begin{bmatrix} {n} \\ {k}\end{bmatrix}_{pq}+q^{n-k+1}\begin{bmatrix} {n} \\ {k-1}\end{bmatrix}_{pq}.$$
(1.4)

Based on this relation, we have

### Lemma 1.1

The $$(p,q)$$-factorial satisfies the following relation:

$$[n+1]_{pq}=p^{2}[n-1]_{pq}+[2]_{pq} \cdot q^{n-1}.$$

### Proof

From relation (1.4) and definition of the $$(p,q)-$$ factorial, for $$k=1$$, we get

\begin{aligned}& \begin{bmatrix} {n+1} \\ {2}\end{bmatrix}_{pq} =p^{2} \begin{bmatrix} {n} \\ {2}\end{bmatrix}_{pq}+q^{n-1}\begin{bmatrix} {n} \\ {1}\end{bmatrix}_{pq}\\& \quad \Rightarrow\quad \frac{[n+1]_{pq}}{[n-1]_{pq}[2]_{pq}}=\frac{p^{2}}{[2]_{pq}} + \frac{q^{n-1}}{[n-1]_{pq}}, \end{aligned}

and we obtain the desired result.â€ƒâ–¡

Some relation related to the p, q-exponential function and p, q-integral are given by the following relations:

$$E_{p,q}(x)=\sum_{n=0}^{\infty} \frac{q^{\binom{n}{2}}x^{n}}{[n]_{p,q}!},$$

$$e_{p,q}(x)E_{p,q} (-x) = 1$$.

$$\int f(x)\,d_{p,q}x=(p-q)x\sum_{k=0}^{\infty }f \biggl( \frac{q^{k}}{p^{k+1}}x \biggr)\frac{q^{k}}{p^{k+1}}.$$

Further, the p, q-Gamma function is given by

$$\Gamma _{p,q}(n)= \int _{0}^{\infty}u^{n-1}E_{p,q}(-qu) \,d_{p,q}u.$$

It is known that the following relation is valid (Proposition 3.3, [26]):

$$\Gamma _{p,q}(x+1)=[x]_{p,q}\Gamma _{p,q}(x),$$
(1.5)

for every x.

In this paper, we introduce modified $$(p,q)$$-Gamma-type operators:

$$G^{(1)}_{n;p,q}(f,x)= \int _{0}^{\infty}K_{n;p,q}(x,u)f \bigl([n]_{p,q}u\bigr)\,d_{p,q}u,$$
(1.6)

with

$$K_{n;p,q}(x,u)=\frac{pqx^{n+3}}{\Gamma _{p,q}(n+3)}E_{p,q} \biggl(- \frac{qx}{u} \biggr)u^{-n-4}.$$
(1.7)

### Remark 1.2

Our operators are a generalization of the operators given in [29]; for $$p\to 1$$, we obtain their class of operators. For $$p\in (0,1)$$ and $$q=0$$, we obtain operators defined in [21].

Now, we give some basic results.

### Lemma 1.3

For $$p,q\in (0, 1)$$ and $$x\in (0,\infty )$$, the operators $$G^{(1)}_{n;p,q}$$ satisfy

$$G^{(1)}_{n;p,q}\bigl(u^{k},x\bigr)= \frac{[n]_{p,q}^{k}x^{k}\Gamma _{p,q}(n+3-k)}{\Gamma _{p,q}(n+3)}= \frac{[n]_{p,q}^{k}x^{k}}{\prod_{j=0}^{k-1}[n+2-j]_{p,q}}.$$

### Proof

By setting $$t=x/u$$, we have

\begin{aligned} G^{(1)}_{n;p,q}\bigl(u^{k},x\bigr) &= \int _{0}^{\infty} \frac{pqx^{n+3}}{\Gamma _{p,q}(n+3)}E_{p,q} \biggl(-\frac{qx}{u} \biggr)u^{-n-4}\bigl([n]_{p,q}u \bigr)^{k}\,d_{p,q}u \\ &=\frac{pq[n]_{p,q}^{k}x^{n+3}}{\Gamma _{p,q}(n+3)} \int _{0}^{\infty}u^{k-n-4}E_{p,q}(-qx/u) \,d_{p,q}u \\ &=\frac{[n]_{p,q}^{k}x^{k}}{\Gamma _{p,q}(n+3)} \int _{0}^{\infty}t^{n+2-k}E_{p,q}(-qt) \,d_{p,q}t \\ &=\frac{[n]_{p,q}^{k}x^{k}\Gamma _{p,q}(n+3-k)}{\Gamma _{p,q}(n+3)}, \end{aligned}

as required.â€ƒâ–¡

As an application of the above Lemma, we have

### Corollary 1.4

For $$p,q\in (0, 1)$$ and $$x\in (0,\infty )$$, the operators $$G^{(1)}_{n;p,q}$$ fulfill

1. (1)

$$G^{(1)}_{n;p,q}(1,x)=1$$,

2. (2)

$$G^{(1)}_{n;p,q}(u,x)=\frac{[n]_{pq}x}{[n+2]_{pq}}$$,

3. (3)

$$G^{(1)}_{n;p,q}(u^{2},x)= \frac{[n]_{pq}^{2}x^{2}}{[n+1]_{pq}[n+2]_{pq}}$$,

4. (4)

$$G^{(1)}_{n;p,q}(u^{3},x)= \frac{[n]_{pq}^{2}x^{3}}{[n+1]_{pq}[n+2]_{pq}}$$,

5. (5)

$$G^{(1)}_{n;p,q}(u^{4},x)= \frac{[n]_{pq}^{3}x^{4}}{[n-1]_{pq}[n+1]_{pq}[n+2]_{pq}}$$.

### Proof

The first one is obvious. For the second, we have:

$$G^{(1)}_{n;p,q}(u,x)= \frac{[n]_{pq}x\Gamma _{pq}(n+2)}{\Gamma _{pq}(n+3)}.$$

From relation (1.5), we obtain

$$G^{(1)}_{n;p,q}(u,x)= \frac{[n]_{pq}x\Gamma _{pq}(n+2)}{\Gamma _{pq}(n+3)}= \frac{[n]_{pq}x\Gamma _{pq}(n+2)}{[n+2]_{pq}\Gamma _{pq}(n+2)}= \frac{[n]_{pq}x}{[n+2]_{pq}}.$$

Similarly, we obtain

\begin{aligned}& G^{(1)}_{n;p,q}\bigl(u^{2},x\bigr)= \frac{[n]_{pq}^{2}x^{2}}{[n+1]_{pq}[n+2]_{pq}},\\& G^{(1)}_{n;p,q}\bigl(u^{3},x\bigr)= \frac{[n]_{pq}^{2}x^{3}}{[n+1]_{pq}[n+2]_{pq}},\\& G^{(1)}_{n;p,q}\bigl(u^{4},x\bigr)= \frac{[n]_{pq}^{3}x^{4}}{[n-1]_{pq}[n+1]_{pq}[n+2]_{pq}}. \end{aligned}

â€ƒâ–¡

As a result of Lemma 1.3 and the linearity of the operator $$G^{(1)}_{n;p,q}$$, we obtain the following:

### Lemma 1.5

For $$p,q\in (0, 1)$$ and $$x\in (0,\infty )$$, the operators $$G^{(1)}_{n;p,q}$$ satisfy

$$G^{(1)}_{n;p,q}\bigl((u-x)^{k},x \bigr)=x^{k}\sum_{j=0}^{k}(-1)^{k-j} \binom{k}{j}\frac{[n]_{p,q}^{j}}{\prod_{i=0}^{j-1}[n+2-i]_{p,q}}.$$

### Lemma 1.6

For $$p,q\in (0, 1)$$ and $$x\in (0,\infty )$$, the operators $$G^{(1)}_{n;p,q}$$ satisfy

1. (1)

$$G^{(1)}_{n;p,q}((u-x),x)= \frac{[n]_{pq}(1-p^{2})-[2]_{pq}q^{n}}{[n+2]_{pq}}x$$,

2. (2)

$$G^{(1)}_{n;p,q}((u-x)^{2},x)= \frac{[n]_{pq}([n]_{pq} +(p^{2}-2)[n+1]_{pq})+[2]_{pq}[n+1]_{pq}q^{n}}{[n+1]_{pq}[n+2]_{pq}}x^{2}$$,

3. (3)

$$G^{(1)}_{n;p,q}((u-x)^{3},x)= \frac{[n]_{pq}(-2[n]_{pq}+(3-p^{2}))[n+1]_{pq})-[2]_{pq}[n+1]_{pq}q^{n}}{[n+1]_{pq}[n+2]_{pq}}x^{3}$$,

4. (4)

$$G^{(1)}_{n;p,q}((u-x)^{4},x)= \frac{[n]_{pq}([n]_{pq}^{2}+2[n]_{pq}[n-1]_{pq}+(p^{2}-4)[n-1]_{pq}[n+1]_{pq})+[2]_{pq}[n-1]_{pq}[n+1]_{pq}q^{n}}{[n-1]_{pq}[n+1]_{pq}[n+2]_{pq}}x^{4}$$.

### Proof

Applying Lemma 1.1 and Lemma 1.5 will give:

1. (1)

$$G^{(1)}_{n;p,q}((u-x),x)=G^{(1)}_{n;p,q}(u,x)-x= \frac{[n]_{pq}x\Gamma _{pq}(n+2)}{\Gamma _{pq}(n+3)}-x= \frac{[n]_{pq}-[n+2]_{pq}}{[n+2]_{pq}}x= \frac{[n]_{pq}-(p^{2}[n]_{pq}+[2]_{pq}q^{n})}{[n+2]_{pq}}x= \frac{[n]_{pq}(1-p^{2})-[2]_{pq}q^{n}}{[n+2]_{pq}}x$$.

2. (2)

Similarly, we obtain: $$G^{(1)}_{n;p,q}((u-x)^{2},x)= \frac{[n]_{pq}^{2}x^{2} -2x^{2}[n]_{pq}[n+1]_{pq}+x^{2}[n+1]_{pq}[n+2]_{pq} }{[n+1]_{pq}[n+2]_{pq}}= \frac{[n]_{pq}([n]_{pq} +(p^{2}-2)[n+1]_{pq})+[2]_{pq}[n+1]_{pq}q^{n}}{[n+1]_{pq}[n+2]_{pq}}x^{2}$$.

3. (3)

$$G^{(1)}_{n;p,q}((u-x)^{4},x)= \frac{[n]_{pq}([n]_{pq}^{2}+2[n]_{pq}[n-1]_{pq}+(p^{2}-4)[n-1]_{pq}[n+1]_{pq})+[2]_{pq}[n-1]_{pq}[n+1]_{pq}q^{n}}{[n-1]_{pq}[n+1]_{pq}[n+2]_{pq}}x^{4}$$.

â€ƒâ–¡

### Remark 1.7

Throughout this paper, we assume that $$({p_{n}})_{n\in \mathbb{N}}$$ and $$({q_{n}})_{n\in \mathbb{N}}$$ are two sequences such that $$0< p_{n},q_{n}<1$$, $$p_{n}\neq q_{n}$$, satisfying $$\lim_{n\rightarrow \infty}p_{n}=\lim_{n\rightarrow \infty}q_{n}=1$$, $$\lim_{n\rightarrow \infty}p_{n}^{n}=\alpha$$ and $$\lim_{n\rightarrow \infty}q_{n}^{n}=\beta$$, where $$0\leq \alpha ,\beta <1$$. Then, from Lemma 1.6, we have

\begin{aligned}& \lim_{n\rightarrow \infty}[n]_{p_{n},q_{n}}G^{(1)}_{n;p_{n},q_{n}} \bigl((u-x),x\bigr) \\& \quad =\lim_{n\to \infty}[n]_{p_{n},q_{n}} \frac{[n]_{p_{n},q_{n}}(1-p_{n}^{2})-[2]_{p_{n},q_{n}}q_{n}^{n}}{[n+2]_{p_{n},q_{n}}}x= (2\alpha -4\beta )x, \\& \lim_{n\rightarrow \infty}[n]_{p_{n},q_{n}}G^{(1)}_{n;p_{n},q_{n}} \bigl((u-x)^{2},x\bigr) \\& \quad = \lim_{n\to \infty}[n]_{p_{n},q_{n}} \frac{[n]_{p_{n},q_{n}}([n]_{p_{n},q_{n}} +(p_{n}^{2}-2)[n+1]_{p_{n},q_{n}})+[2]_{p_{n},q_{n}}[n+1]_{p_{n},q_{n}}q_{n}^{n}}{[n+1]_{p_{n},q_{n}}[n+2]_{p_{n},q_{n}}}x^{2} \\& \quad =2\alpha x^{2}, \\& \lim_{n\to \infty}[n]_{pq}G^{(1)}_{n;p,q} \bigl((u-x)^{3},x\bigr) \\& \quad =\lim_{n\to \infty}[n]_{pq} \frac{[n]_{pq}(-2[n]_{pq}+(3-p^{2}))[n+1]_{pq})-[2]_{pq}[n+1]_{pq}q^{n}}{[n+1]_{pq}[n+2]_{pq}}x^{3}=(2 \alpha -4\beta )x^{3}, \\& \lim_{n\rightarrow \infty}[n]_{p_{n},q_{n}}G^{(1)}_{n;p_{n},q_{n}} \bigl((u-x)^{4},x\bigr) \\& \quad = \lim_{n\to \infty}[n]_{p_{n},q_{n}} \\& \qquad {}\times \frac{[n]_{pq}([n]_{pq}^{2}+2[n]_{pq}[n-1]_{pq}+(p^{2}-4)[n-1]_{pq}[n+1]_{pq})+[2]_{pq}[n-1]_{pq}[n+1]_{pq}q^{n}}{[n-1]_{pq}[n+1]_{pq}[n+2]_{pq}}x^{4} \\& \quad =2\alpha x^{4}. \end{aligned}

Next results prove the Korovkin-type theorem for the $$G^{(1)}_{n;p,q}$$. The Korovkin-type theorem and its versions are widely studied; see, for example, [2â€“9, 17, 20, 23].

### Theorem 1.8

Let $$G^{(1)}_{n;p,q}$$ be a sequence of positive linear operators defined on $$C[0,\infty )$$, such that for every $$i\in \{0,1,2\}$$,

$$\lim_{n\to \infty}{ \bigl\Vert G^{(1)}_{n;p_{n},q_{n}}(e_{i};x)-e_{i} \bigr\Vert }=0,$$
(1.8)

where $$e_{i}=x^{i}$$. Then, for every $$f\in C[0,\infty )$$,

$$\lim_{n\to \infty}{ \bigl\Vert G^{(1)}_{n;p_{n},q_{n}}(f;x)-f \bigr\Vert }=0,$$
(1.9)

uniformly for every $$x\in [a,b]\subset [0,\infty )$$.

### Proof

From Corollary 1.4, we have

\begin{aligned} \bigl\Vert G^{(1)}_{n;p,q}(e_{0};x)-e_{0} \bigr\Vert =1-1=0, \end{aligned}
\begin{aligned} \bigl\Vert G^{(1)}_{n;p_{n},q_{n}}(e_{1};x)-e_{1} \bigr\Vert = \biggl\Vert \frac{[n]_{p_{n}q_{n}}x}{[n+2]_{p_{n}q_{n}}}-x \biggr\Vert =0 \end{aligned}

and

\begin{aligned} & \bigl\Vert G^{(1)}_{n;p_{n},q_{n}}(e_{2};x)-e_{2} \bigr\Vert \\ &\quad = \biggl\Vert \frac{[n]_{p_{n}q_{n}}^{2}x^{2}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}-x^{2} \biggr\Vert =0. \end{aligned}

The proof of theorem follows from the Korovkin theorem [1].â€ƒâ–¡

## 2 Some direct results

With $$B[0,\infty )$$, $$C[0,\infty )$$ and $$C_{B}([0,\infty ))$$, we will denote the space of all bounded functions, continuous functions, and continuous, bounded functions defined in the interval $$[0,\infty )$$. Let be given $$\eta >0$$, then the Petree K-functional [28] is defined as follows:

$$K(t,\eta )=\inf_{r\in C_{B}^{2}([0,\infty ))}\bigl\{ \Vert t-r \Vert +\eta \bigl\Vert r^{\prime \prime } \bigr\Vert \bigr\} ,$$

and $$C_{B}^{2}([0,\infty ))=\{r/r^{\prime },r^{\prime \prime }\in C_{B}([0,\infty )) \}$$, with the norm

$$\Vert t \Vert _{C_{B}^{2}}= \Vert t \Vert _{\infty}+ \bigl\Vert t^{\prime} \bigr\Vert _{\infty} + \bigl\Vert t^{ \prime \prime} \bigr\Vert _{\infty}.$$

It is proven in [14] and [15] that exists a constant $$C>0$$ such that

$$K(t,\eta )\leq C\cdot \omega _{2}(t,\sqrt{\eta}),$$
(2.1)

where

$$\omega _{2}(t,\eta )=\sup_{0< \vert h \vert \leq \eta}\sup _{u, u+\eta \in [0, \infty )} \bigl\vert t(u+2h)-2t(u+h)+t(u) \bigr\vert .$$

### Theorem 2.1

If $$t\in C_{B}[0,\infty )$$, then

\begin{aligned} & \bigl\Vert G^{(1)}_{n;p_{n},q_{n}}t-t \bigr\Vert \\ & \quad \leqq \omega (t;\sqrt{n}) \\ &\qquad {}\times \biggl(1+\frac{1}{\sqrt{n}} \biggl[ \frac{[n]_{p_{n}q_{n}}([n]_{p_{n}q_{n}} +(p_{n}^{2}-2)[n+1]_{p_{n}q_{n}})+[2]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}q_{n}^{n}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{2} \biggr]^{\frac{1}{2}} \biggr). \end{aligned}

### Proof

From properties of the modulus of continuity and fact that operators $$G^{(1)}_{n;p_{n},q_{n}}$$ are positive and linear, for any $$t\in C_{B}[0,\infty )$$, we obtain

\begin{aligned} \bigl\vert G^{(1)}_{n;p_{n},q_{n}}(t;y)-t(y) \bigr\vert &\leqq \int _{0}^{\infty}K_{n;p_{n},q_{n}}(y,u) \bigl\vert t\bigl([n]_{p_{n},q_{n}}u\bigr)-t(y) \bigr\vert \,d_{p_{n},q_{n}}u \\ &\leqq \omega (t;\eta ) \biggl(1+ \int _{0}^{\infty}K_{n;p_{n},q_{n}}(y,u) \frac{ \vert [n]_{p_{n},q_{n}}u-y \vert }{\eta}\,d_{p_{n},q_{n}}u \biggr). \end{aligned}
(2.2)

Let us set

$$B:=\frac{1}{\eta} \int _{0}^{\infty}K_{n;p_{n},q_{n}}(y,u) \bigl\vert [n]_{p_{n},q_{n}}u-y \bigr\vert \,d_{p_{n},q_{n}}u.$$

Then, using the Cauchyâ€“Schwarz inequality, we get

\begin{aligned} B&\leqq \biggl[ \int _{0}^{\infty}K_{n;p_{n},q_{n}}(y,u) \,d_{p_{n},q_{n}}u \biggr]^{\frac{1}{2}}\cdot \biggl[ \int _{0}^{\infty}K_{n;p_{n},q_{n}}(y,u) \bigl\vert [n]_{p_{n},q_{n}}u-y \bigr\vert ^{2} \,d_{p_{n},q_{n}}u \biggr]^{\frac{1}{2}} \\ &=\bigl[G^{(1)}_{n,p_{n},q_{n}}\bigl((s-y)^{2},y\bigr) \bigr]^{\frac{1}{2}} \\ & = \biggl[ \frac{[n]_{p_{n}q_{n}}([n]_{p_{n}q_{n}} +(p_{n}^{2}-2)[n+1]_{p_{n}q_{n}})+[2]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}q_{n}^{n}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{2} \biggr]^{\frac{1}{2}}. \end{aligned}
(2.3)

Putting $$\eta =\sqrt{n}$$, we get the result.â€ƒâ–¡

Next result gives an upper bound for $$G^{(1)}_{n,p_{n},q_{n}}$$-Gamma operators.

### Theorem 2.2

For any $$g \in C_{B}[0,\infty )$$,

$$\bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y) \bigr\vert \leqq \Vert g \Vert _{C}.$$

### Proof

From the definition of the modified $$(p,q)$$-Gamma-type operators in (1.6), we have

\begin{aligned} \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y) \bigr\vert &\leqq \sup _{s\in \mathbb{R}^{+}}{ \bigl\vert g(s) \bigr\vert } \cdot \int _{0}^{\infty} \bigl\vert K_{n;p_{n},q_{n}}(y,u) \bigr\vert \,d_{p_{n},q_{n}}u= \Vert g \Vert _{C}. \end{aligned}

â€ƒâ–¡

### Theorem 2.3

For $$y \in (0,\infty )$$, $$g\in C_{B}[0,\infty )$$, there exists a $$M\in {\mathbb{R}}^{+}$$, such that

$$\bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g,y)-g(y) \bigr\vert \leq M \omega _{2}\bigl(g,\sqrt{ \bigl\vert J(y) \bigr\vert +I^{2}(y)}\bigr)+ \omega \bigl(g, \bigl\vert I(y) \bigr\vert \bigr),$$

where $$I(y)= \frac{[n]_{p_{n}q_{n}}(1-p_{n}^{2})-[2]_{p_{n}q_{n}}q_{n}^{n}}{[n+2]_{p_{n}q_{n}}}y$$ and

$$J(y)= \frac{[n]_{p_{n}q_{n}}([n]_{p_{n}q_{n}} +(p_{n}^{2}-2)[n+1]_{p_{n}q_{n}})+[2]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}q_{n}^{n}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{2}.$$

### Proof

For any $$y\in (0,\infty )$$, we denote by

$$G^{(2)} (n,p_{n},q_{n}) (g,y)=G^{(1)}_{n,p_{n},q_{n}}(g,y)+g(y)-g\bigl(I(y)+y\bigr).$$

Then, from Lemma (1.5), we obtain

$$G^{(2)}_{n,p_{n},q_{n}}\bigl((s-y),y\bigr)=G_{n,p_{n},q_{n}} \bigl((s-y),y\bigr)+(s-y)-\bigl(I(y)+y-y\bigr)=I(y)-I(y)=0.$$

Let $$y,s\in (0,\infty )$$ and $$r(y)\in C_{B}^{2}([0,\infty ))$$. Using the Taylor formula, we get:

$$r(s)=r(y)+r^{\prime }(y) (s-y)+ \int _{y}^{s}\bigl(r^{\prime \prime }(v) (s-v) \bigr)\,dv,$$

and it yields

\begin{aligned} & \bigl\vert G^{(2)}_{n,p_{n},q_{n}}(r,y)-r(y) \bigr\vert \\ &\quad= \biggl\vert r^{\prime }(y)G_{n,p_{n},q_{n}}^{(2)} \bigl((s-y),y\bigr)+G^{(2)}_{n,p_{n},q_{n}} \biggl( \int _{y}^{s}\bigl(r^{\prime \prime }(v) (s-v) \bigr)\,dv,y \biggr) \biggr\vert \\ &\quad= \biggl\vert G^{(2)}_{n,p_{n},q_{n}} \biggl( \int _{y}^{s}\bigl(r^{\prime \prime }(v) (s-v) \bigr)\,dv,y \biggr) \biggr\vert \\ &\quad= \biggl\vert G^{(2)}_{n,p_{n},q_{n}} \biggl( \int _{y}^{s}\bigl(r^{\prime \prime }(v) (s-v) \bigr)\,dv,y \biggr)- \int _{y}^{I(y)+y}{r^{\prime \prime }(v) \bigl(I(y)+y-v\bigr)}\,dv \biggr\vert \\ &\quad\leq G^{(1)}_{n,p_{n},q_{n}} \biggl( \int _{y}^{s} \bigl\vert r^{\prime \prime }(v) \bigr\vert (s-v)\,dv,y \biggr)+ \int _{y}^{I(y)+y}{ \bigl\vert r^{\prime \prime }(v) \bigr\vert \bigl\vert \bigl(I(y)+y-v\bigr) \bigr\vert }\,dv \\ &\quad\leq \bigl( \bigl\vert J(y) \bigr\vert +I^{2}(y)\bigr) \bigl\Vert r^{\prime \prime } \bigr\Vert . \end{aligned}

From Theorem 2.2, we have that $$|G^{(1)}_{n,p_{n},q_{n}}(g,y)|\leq \|f\|$$, then

\begin{aligned} & \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g,y)-g(y) \bigr\vert \\ &\quad= \bigl\vert G^{(2)} (n,p_{n},q_{n}) (g,y)+g\bigl(I(y)+y\bigr)-2g(y) \bigr\vert \\ &\quad\leq \bigl\vert G^{(2)} (n,p_{n},q_{n}) (g-r,y)-(g-r)y \bigr\vert \\ &\qquad {}+ \bigl\vert G^{(2)} (n,p_{n},q_{n}) (r,y)-r(y) \bigr\vert + \bigl\vert g\bigl(I(y)+y\bigr)-g(y) \bigr\vert \\ &\quad\leq 4 \Vert g-r \Vert +\bigl( \bigl\vert J(y) \bigr\vert +I^{2}(y)\bigr) \bigl\Vert r^{\prime \prime } \bigr\Vert +\omega \bigl(g, \bigl\vert I(y) \bigr\vert \bigr). \end{aligned}

Taking infimum for all $$r\in C_{B}^{2}([0,\infty ))$$ and relation (2.1), we obtain our result.â€ƒâ–¡

In [15], the following modulus are given:

\begin{aligned} \omega _{\gamma}(g;\eta ):=\sup_{0< \vert h \vert \leqq \eta} ~\sup _{y,y+ h \gamma (y) \in [0,\infty )} \bigl\{ \bigl\vert g \bigl(y+h\gamma (y) \bigr)-g(y) \bigr\vert \bigr\} \end{aligned}

and

\begin{aligned} \omega _{2}^{\rho}(g;\eta ):=\sup_{0< \vert h \vert \leqq \eta} ~\sup_{y,y\pm h \rho (y)\in [0,\infty )} \bigl\{ \bigl\vert g \bigl(y+h\rho (y) \bigr) -2g(y)+g \bigl(y-h\rho (y) \bigr) \bigr\vert \bigr\} , \end{aligned}

$$\rho (y)=\sqrt{(y-a)(b-y)}$$, and K-functional:

\begin{aligned} K_{2, \rho (y)}(g,\eta )=\inf_{r \in W^{2}(\rho )} \bigl\{ \Vert g-r \Vert _{C[0, \infty )}+\eta \bigl\Vert \rho ^{2}r'' \bigr\Vert _{C[0,\infty )} \bigr\} , \end{aligned}

where $$\eta >0$$.

$$W^{2}(\rho )=\bigl\{ r\in C_{B}[0,\infty ):r' \in AC[0,\infty ), ~ \rho ^{2}r'' \in C_{B}[0,\infty )\bigr\} \quad \text{and}\quad r' \in AC[0,\infty ).$$

### Theorem 2.4

Let $$\rho =\sqrt{y(1-y)}$$, $$g\in C_{B}[0,1]$$ and $$y\in [0,1]$$, $$n\in \mathbb{N}$$. Then,

\begin{aligned} \bigl\Vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\Vert \leqq{}& 4K_{2, \rho (y)} \biggl(g, \frac{G^{(1)}_{n,p_{n},q_{n}} ((s-y)^{2};y )+\alpha _{1}(n,p_{n},q_{n})}{4\rho ^{2}(y)} \biggr) \\ &{} +\omega _{\gamma} \biggl(g; \frac{\alpha _{1}(n,p_{n},q_{n})}{ \gamma (y)} \biggr), \end{aligned}

where $$\alpha _{1}(n,p_{n},q_{n})= \frac{[n]_{p_{n}q_{n}}}{[n+2]_{p_{n}q_{n}}}$$.

### Proof

Let

$$G^{(3)}_{n,p_{n},q_{n}}(g;y)=G^{(1)}_{n,p_{n},q_{n}}(g;y)+g(y)- g \bigl(y+\beta _{1}(n,p_{n},q_{n},y) \bigr),$$

where

$$\beta _{1}(n,p_{n},q_{n},y) = \frac{[n]_{p_{n}q_{n}}(1-p_{n}^{2})-[2]_{p_{n}q_{n}}q_{n}^{n}}{[n+2]_{p_{n}q_{n}}}y.$$

Then,

$$G^{(3)}_{n,p_{n},q_{n}}(1;y)=1\quad \text{and} \quad G^{(3)}_{n,p_{n},q_{n}} \bigl((s-y); y \bigr)=0 .$$

Let $$r \in W^{2}(\rho )$$. Using the Taylor formula, we obtain

$$r(s)=r(y)+r'(y) (s-y)+ \int _{y}^{s}(s-v)r''(v) \,{\mathrm{d}}v \quad \bigl(s \in [0,\infty \bigr) ),$$

and

\begin{aligned} G^{(3)}_{n,p_{n},q_{n}}(r;y)-r(y) ={}&G^{(1)}_{n,p_{n},q_{n}} \biggl( \int _{y}^{s}(s-v)r''(v) \,{\mathrm{d}}v; y \biggr) \\ & {}- \int _{y}^{y+\beta _{1}(n,p_{n},q_{n},y)} \bigl[y+\beta _{1}(n,p_{n},q_{n},y)-v\bigr]r''(v) \, {\mathrm{d}}v. \end{aligned}

Therefore, we have

\begin{aligned} & \bigl\vert G^{(3)}_{n,p_{n},q_{n}}(r;y)-r(y) \bigr\vert \\ &\quad \leqq G^{(1)}_{n,p_{n},q_{n}} \biggl( \biggl\vert \int _{y}^{s}(s-v)r''(v) \,{\mathrm{d}}v \biggr\vert ;y \biggr) \\ &\qquad{} + \int _{y}^{y+\beta _{1}(n,p_{n},q_{n},y)} \bigl\vert y+\beta _{1}(n,p_{n},q_{n},y)-v \bigr\vert \cdot \bigl\vert r''(v) \bigr\vert \,{\mathrm{d}}v \\ &\quad \leqq \biggl\Vert \rho ^{2}r''(y)~G^{(1)}_{n,p_{n},q_{n}} \biggl( \biggl\vert \int _{y}^{s}\frac{ \vert s-v \vert }{\rho ^{2}(v)}\,{ \mathrm{d}}v \biggr\vert ; y \biggr) + \bigl\Vert \rho ^{2}r''(y) \bigr\Vert \biggr\Vert \\ &\qquad{} \cdot \biggl\vert \int _{y}^{y+\beta _{1}(n,p_{n},q_{n},y)} \frac{ \vert y+\beta _{1}(n,p_{n},q_{n},y)-v \vert }{\rho ^{2}(v)}\,{ \mathrm{d}}v \biggr\vert . \end{aligned}

For $$v=\nu y+(1-\nu )s$$ $$(\nu \in [0,1])$$. Since $$\rho ^{2}$$ is concave on $$[0,\infty )$$, it follows that $$\rho ^{2}(v)\ge \nu \rho ^{2}(y)+(1-\nu )\rho ^{2}(s)$$ and hence

\begin{aligned} \frac{ \vert s-v \vert }{\rho ^{2}(v)}=\frac{\nu \vert y-s \vert }{\rho ^{2}(v)} \leqq \frac{\nu \vert y-s \vert }{\nu \rho ^{2}(y)+(1-\nu )\rho ^{2}(s)} \leqq \frac{ \vert y-s \vert }{\rho ^{2}(y)}. \end{aligned}

Thus, we have

$$\bigl\Vert G^{(3)}_{n,p_{n},q_{n}}(r)-r \bigr\Vert \leqq \frac{ \Vert \rho ^{2}r'' \Vert _{C[0,\infty )}}{\rho ^{2}(y)} \bigl\{ \bigl[G^{(1)}_{n,p_{n},q_{n}} \bigl((s-y)^{2};y \bigr) \bigr] +y\beta _{1}(n,p_{n},q_{n},y) \bigr\} .$$

From the above relations, we obtain

\begin{aligned} & \bigl\Vert G^{(3)}_{n,p_{n},q_{n}}(g,y)-g(y) \bigr\Vert \\ &\quad \leqq \bigl\Vert G^{(3)}_{n,p_{n},q_{n}}(g-r) \bigr\Vert + \bigl\Vert G^{(3)}_{n,p_{n},q_{n}}(r)-r \bigr\Vert + \Vert g-r \Vert + \bigl\Vert g \bigl(y+\beta _{1}(n,p_{n},q_{n},y) \bigr)-g(y) \bigr\Vert \\ &\quad \leqq 4 \Vert g-r \Vert +\frac{ \Vert \rho ^{2}r'' \Vert }{ \rho ^{2}(y)} \bigl[G^{(1)}_{n,p_{n},q_{n}} \bigl((s-y)^{2};y \bigr) +y \beta _{1}(n,p_{n},q_{n},y) \bigr] \\ &\qquad {}+ \bigl\Vert g\bigl(y+\beta _{1}(n,p_{n},q_{n},y) \bigr)-g(y) \bigr\Vert . \end{aligned}

On the other hand,

\begin{aligned} \bigl\Vert g\bigl(y+\beta _{1}(n,p_{n},q_{n},y) \bigr)-g(y) \bigr\Vert &\leqq \biggl\Vert g \biggl(y+ \gamma (y) \frac{G^{(1)}_{n,p_{n},q_{n}} ((s-y);y )}{ \gamma (y)} \biggr) -g(y) \biggr\Vert \\ &\leqq \omega _{\gamma} \biggl(g; \frac{\beta _{1}(n,p_{n},q_{n},y)}{ \gamma (y)} \biggr). \end{aligned}

Therefore,

\begin{aligned} \bigl\Vert G^{(1)}_{n,p_{n},q_{n}}(g, y)-g(y) \bigr\Vert \leqq {}& 4 K_{2,\rho (y)} \biggl(g, \frac{G^{(1)}_{n,p_{n},q_{n}} ((s-y)^{2};y )+y\beta _{1}(n,p_{n},q_{n},y)}{4\rho ^{2}(y)} \biggr) \\ &{} + \omega _{\gamma} \biggl(g; \frac{\beta _{1}(n,p_{n},q_{n},y)}{ \gamma (y)} \biggr). \end{aligned}
(2.4)

From inequality

1. (1)
$$\frac{[n]_{p_{n}q_{n}}(1-p_{n}^{2})-[2]_{p_{n}q_{n}}q_{n}^{n}}{[n+2]_{p_{n}q_{n}}}y \leqq \frac{[n]_{p_{n}q_{n}}}{[n+2]_{p_{n}q_{n}}}.$$

It follows from Theorem 2.4

\begin{aligned} &K_{2,\rho (y)} \biggl(g, \frac{G^{(1)}_{n,p_{n},q_{n}} ((s-y)^{2};y )+y\beta _{1}(n,p_{n},q_{n},y)}{4\rho ^{2}(y)} \biggr) \\ &\quad \leqq K_{2,\rho (y)} \biggl(g, \frac{G^{(1)}_{n,p_{n},q_{n}} ((s-y)^{2};y )+\alpha _{1}(n,p_{n},q_{n})}{4\rho ^{2}(y)} \biggr), \end{aligned}
2. (2)
$$\omega _{\gamma} \biggl(g; \frac{\beta _{1}(n,p_{n},q_{n},y)}{ \gamma (y)} \biggr)\leqq \omega _{\gamma} \biggl(g; \frac{\alpha _{1}(n,p_{n},q_{n})}{\gamma (y)} \biggr)$$

$$\forall y\in [0,1]$$. Finally, we have

\begin{aligned} \bigl\Vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\Vert \leqq {}&4 K_{2,\rho (y)} \biggl(g, \frac{G^{(1)}_{n,p_{n},q_{n}} ((s-y)^{2};y )+\alpha _{1}(n,p_{n},q_{n})}{4\rho ^{2}(y)} \biggr) \\ &{} +\omega _{\gamma} \biggl(g; \frac{\alpha _{1}(n,p_{n},q_{n})}{\gamma (y)} \biggr), \end{aligned}

as asserted by the theorem.â€ƒâ–¡

### Theorem 2.5

Let $$g\in C[0,N]$$, N is a finite number. Then,

$$\bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert \leqq \frac{2}{N} \Vert g \Vert c^{2}+ \frac{3}{4} \bigl(N+c^{2}+2\bigr)\omega _{2}(g;c),$$

where

$$c=\sqrt[4]{G^{(1)}_{n,p_{n},q_{n}}{ \bigl((s-y)^{2};y \bigr)}}.$$

### Proof

Let $$g_{S}$$ be the Steklov function of the second order for $$g(y)$$. We know that

$$G^{(1)}_{n,p_{n},q_{n}}(e_{0};y)=1,$$

which follows from Corollary (1.4), and

\begin{aligned} \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert &\leqq \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g-g_{S};y) \bigr\vert + \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g_{S};y)-y_{S}(y) \bigr\vert + \bigl\vert g_{S}(y)-g(y) \bigr\vert \\ &\leqq 2 \Vert g_{S}-g \Vert + \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g_{S};y)-g_{S}(y) \bigr\vert . \end{aligned}
(2.5)

It follows from Lemmas in [30]

$$\bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert \leqq \frac{3}{2}\omega _{2}(g;c) + \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g_{S};y)-g_{S}(y) \bigr\vert .$$
(2.6)

As $$g_{S}\in C^{2}[0,N]$$, and Lemmas in [17], we get

$$\bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g_{S};y)-g_{S}(y) \bigr\vert \leqq \bigl\Vert g_{S}^{ \prime} \bigr\Vert \;\sqrt{G^{(1)}_{n,p_{n},q_{n}}{ \bigl((s-y)^{2};y \bigr)}}+ \frac{1}{2} \bigl\Vert g_{S}^{\prime \prime} \bigr\Vert G^{(1)}_{n,p_{n},q_{n}}{ \bigl((s-y)^{2};y \bigr)}.$$

The following inequality is valid [30]:

$$\bigl\Vert g_{S}^{\prime \prime} \bigr\Vert \leqq \frac{3}{2c^{2}}\omega _{2}(g;c).$$
(2.7)

In the light of (2.6) and (2.7), we obtain:

$$\bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g_{S};y)-g_{S}(y) \bigr\vert \leqq \bigl\Vert g_{S}^{ \prime} \bigr\Vert \;\sqrt{G^{(1)}_{n,p_{n},q_{n}}{ \bigl((s-y)^{2};y \bigr)}}+ \frac{3}{4c^{2}}\omega _{2}(g;c) G^{(1)}_{n,p_{n},q_{n}}{ \bigl((s-y)^{2};y \bigr)}.$$

From relation (2.7) and the Landau inequality [22], we get

$$\bigl\Vert g_{S}^{\prime} \bigr\Vert \leqq \frac{2}{N} \Vert g \Vert +\frac{3N}{4c^{2}}\omega _{2}(g;c).$$
(2.8)

Using relations (2.7) and (2.8) and upon setting

$$c=\sqrt[4]{G^{(1)}_{n,p_{n},q_{n}}{ \bigl((s-y)^{2};y \bigr)}},$$

we obtain

$$\bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g_{S};y)-g_{S}(y) \bigr\vert \leqq \frac{2}{N} \Vert g \Vert c^{2}+ \frac{3}{4}\bigl(N+c^{2}\bigr)\omega _{2}(g;c).$$

The proof of the theorem follows from relation (2.6).â€ƒâ–¡

### Theorem 2.6

Let $$g\in C_{B}[0,\infty )$$. Then,

$$\bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert \leqq D(n,p_{n},q_{n},y) \Vert g \Vert _{C_{B}^{2}},$$

for $$y\geqq 0$$, where

\begin{aligned} D(n,p_{n},q_{n}, y) ={}& \biggl[ \frac{[n]_{p_{n}q_{n}}(1-p_{n}^{2})-[2]_{p_{n}q_{n}}q_{n}^{n}}{[n+2]_{p_{n}q_{n}}}y \biggr] \\ &{} + \biggl[ \frac{[n]_{p_{n}q_{n}}([n]_{p_{n}q_{n}} +(p_{n}^{2}-2)[n+1]_{p_{n}q_{n}})+[2]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}q_{n}^{n}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{2} \biggr]. \end{aligned}

### Proof

From the Taylor formula, it follows

$$G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) =G^{(1)}_{n,p_{n},q_{n}} \bigl((s-y);y \bigr)g^{\prime}(y) +\frac{1}{2}G^{(1)}_{n,p_{n},q_{n}} \bigl((s-y)^{2};y \bigr)g^{\prime \prime}(\iota ),$$

where $$\iota \in (y,s)$$. From the above relation, we have

\begin{aligned} & \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert \\ &\quad= \bigl\Vert g^{\prime} \bigr\Vert \cdot \biggl[ \frac{[n]_{p_{n}q_{n}}(1-p_{n}^{2})-[2]_{p_{n}q_{n}}q_{n}^{n}}{[n+2]_{p_{n}q_{n}}}y \biggr] \\ &\qquad{}+\frac{ \Vert g^{\prime \prime} \Vert }{2} \biggl[ \frac{[n]_{p_{n}q_{n}}([n]_{p_{n}q_{n}} +(p_{n}^{2}-2)[n+1]_{p_{n}q_{n}})+[2]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}q_{n}^{n}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{2} \biggr] \\ &\quad\leqq D(n,p_{n},q_{n}, y) \Vert g \Vert _{C_{B}^{2}}. \end{aligned}

â€ƒâ–¡

### Theorem 2.7

Let $$g\in C[0,\infty )$$. Then,

\begin{aligned} \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert &\leqq 2 \mathcal{M} \biggl[ \omega _{2} \biggl(g;\sqrt{\frac{1}{2} \; D(n,p_{n},q_{n},y)} \biggr) \\ &\quad +\min \biggl\{ 1,\frac{1}{2}D(n,p_{n},q_{n},y) \biggr\} \Vert g \Vert _{\infty} \biggr], \end{aligned}

where $$\mathcal{M}>0$$ is a constant, and $$D(n,p_{n},q_{n},y)$$ is as in Theorem 2.6.

### Proof

Let

$$g(t)-g(y)=g(t)-r(t)+r(t)-r(y)+r(y)-g(y),$$

then

\begin{aligned} \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert &\leqq \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g-r;y) \bigr\vert \\ &\quad + \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(r;y)-r(y) \bigr\vert + \bigl\vert g(y)-r(y) \bigr\vert . \end{aligned}

Considering that $$g\in C_{B}^{2}$$ and Theorems 2.2 and 2.6, we get

\begin{aligned} \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert &\leqq 2 \Vert g-r \Vert + D(n,p_{n},q_{n}, y) \Vert r \Vert _{C_{B}^{2}} \\ &=2 K \biggl(g;\frac{1}{2}D(n,p_{n},q_{n},y) \biggr). \end{aligned}

The following relation is valid [15]

$$K(g;\eta )\leqq L \Bigl[\omega _{2}(g;\sqrt{\eta}) +\min \{1,\eta \} \Vert g \Vert _{\infty} \Bigr],$$

for $$\forall \eta >0$$, and $$L>0$$ is a positive constant. The proof of the theorem follows from the last two relations.â€ƒâ–¡

The next result gives an estimation of $$G^{(1)}_{n,p_{n},q_{n}}$$-operators in Lipschitz space $${\mathrm{Lip}}_{L}{\gamma}$$ [27] given by the relation:

\begin{aligned} {\mathrm{Lip}}_{L}(\gamma ):= \biggl\{ g \in C_{B}[0, \infty ): \bigl\vert g(s)-g(y) \bigr\vert \leqq {L}\frac{ \vert s-y \vert ^{\gamma}}{(y+s)^{\frac{\gamma}{2}}},\; y \in (0, \infty )\; s\in (0,\infty ) \biggr\} , \end{aligned}

$$L>0$$ is a constant, $$\gamma \in (0, 1]$$.

### Theorem 2.8

Let $$g\in {\mathrm{Lip}}_{L}(\gamma )$$. Then, $$\forall y, t \in (0,\infty )$$, $$n\in \mathbb{N}$$ and $$\gamma \in (0, 1]$$,

\begin{aligned} & \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert \\ &\quad \leqq \frac{T}{ (y+t)^{\frac{\gamma}{2}}} \biggl(\frac{ L}{(y+t)^{\frac{\gamma}{2}}} \\ &\qquad {}\times \biggl\{ \frac{[n]_{p_{n}q_{n}}([n]_{p_{n}q_{n}} +(p_{n}^{2}-2)[n+1]_{p_{n}q_{n}})+[2]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}q_{n}^{n}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{2} \biggr\} ^{\frac{\gamma}{2}} \biggr)^{\frac{\gamma}{2}}, \end{aligned}

$$T>0$$ is a constant.

### Proof

Let $$g \in {\mathrm{Lip}}_{L}^{*}(\gamma )$$ and $$\gamma \in (0, 1]$$. Then,

I. For $$\gamma =1$$, we have

\begin{aligned} & \bigl\vert G^{(1)}_{n,p_{n},q_{n}} \bigl(g(s);y \bigr)-g(y) \bigr\vert \\ &\quad \leqq \bigl\vert G^{(1)}_{n,p_{n},q_{n}} \bigl( \bigl\vert g(s)-g(y) \bigr\vert ;y \bigr) \bigr\vert \\ &\quad \leqq {T}\cdot G^{(1)}_{n,p_{n},q_{n}} \biggl( \frac{ \vert s-y \vert }{(y+s)^{\frac{1}{2}}};y \biggr) \\ &\quad \leqq \frac{T}{(y+s)^{\frac{1}{2}}}G^{(1)}_{n,p_{n},q_{n}} \bigl( \vert s-y \vert ;y \bigr) \end{aligned}

for $$T>0$$ constant.

Using the Cauchyâ€“Schwarz inequality, we have

\begin{aligned} & \bigl\vert G^{(1)}_{n,p_{n},q_{n}} \bigl(g(s);y \bigr)-g(y) \bigr\vert \\ &\quad \leqq \frac{{T}}{(y+s)^{\frac{1}{2}}}\; G^{(1)}_{n,p_{n},q_{n}} \bigl( \vert s-y \vert ;y \bigr) \\ &\quad \leqq \frac{{T}}{(y+s)^{\frac{1}{2}}}\; \sqrt{G^{(1)}_{n,p_{n},q_{n}} \bigl((s-y)^{2};y \bigr)} \\ &\quad =\frac{{T}}{(y+s)^{\frac{1}{2}}}\; \biggl( \frac{[n]_{p_{n}q_{n}}([n]_{p_{n}q_{n}} +(p_{n}^{2}-2)[n+1]_{p_{n}q_{n}})+[2]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}q_{n}^{n}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{2} \biggr)^{\frac{1}{2}}. \end{aligned}

II. For $$\gamma \in (0,1)$$, we have

\begin{aligned} & \bigl\vert G^{(1)}_{n,p_{n},q_{n}} \bigl(g(s);y \bigr)-g(y) \bigr\vert \\ &\quad \leqq \bigl\vert G^{(1)}_{n,p_{n},q_{n}} \bigl( \bigl\vert g(s)-g(y) \bigr\vert ;y \bigr) \bigr\vert \\ &\quad \leqq {T}\cdot G^{(1)}_{n,p_{n},q_{n}} \biggl( \frac{ \vert s-y \vert ^{\gamma}}{ (y+s)^{\frac{\gamma}{2}}};y \biggr) \\ &\quad \leqq \frac{T}{(y+s)^{\frac{\gamma}{2}}}\;G^{(1)}_{n,p_{n},q_{n}} \bigl( \vert s-y \vert ^{\gamma};y \bigr). \end{aligned}

From the HÃ¶lder inequality under the following conditions

$$\frac{1}{\gamma}, \frac{1}{1-\gamma},$$

it follows

$$\bigl\vert G^{(1)}_{n,p_{n},q_{n}} \bigl(g(s);y \bigr)-g(y) \bigr\vert \leqq \frac{{T}}{(y+s)^{\frac{\gamma}{2}}} \bigl[G^{(1)}_{n,p_{n},q_{n}} \bigl( \vert s-y \vert ;y \bigr) \bigr]^{\gamma}$$

for $$T>0$$ constant. Applying the Cauchyâ€“Schwarz inequality, we have:

\begin{aligned} & \bigl\vert G^{(1)}_{n,p_{n},q_{n}} \bigl(g(s);y \bigr)-g(y) \bigr\vert \\ &\quad \leqq \frac{ T}{(y+s)^{\frac{\gamma}{2}}} \bigl[\sqrt{G^{(1)}_{n,p_{n},q_{n}} \bigl((s-y)^{2};y \bigr)} \bigr]^{\gamma} \\ &\quad =\frac{ T}{(y+s)^{\frac{\gamma}{2}}} \biggl\{ \frac{[n]_{p_{n}q_{n}}([n]_{p_{n}q_{n}} +(p_{n}^{2}-2)[n+1]_{p_{n}q_{n}})+[2]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}q_{n}^{n}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{2} \biggr\} ^{\frac{\gamma}{2}}. \end{aligned}

â€ƒâ–¡

## 3 Weighted approximation

Let $$\zeta (y)=y^{2}+1$$ be the weight function. We denote by $$B_{\zeta}[0,\infty )$$, $$C_{\zeta}[0,\infty )$$ and $$C^{*}_{\zeta}[0,\infty )$$ the space of functions g defined on $$[0,\infty )$$ and satisfying, respectively: $$|g(y)|\leqq T_{g} \zeta (y)$$, where $$T_{g}$$ is a constant, space of all continuous functions and subspace of $$C_{\zeta}[0,\infty )$$ for which $$\frac{g(y)}{\zeta (y)}$$ is convergent as $$y\to \infty$$.

The space $$B_{\zeta}[0, \infty )$$ is a normed linear space defined by the norm as follows:

$$\Vert g \Vert _{\zeta}=\sup_{y\geqq 0} \frac{ \vert g(y) \vert }{\zeta (y)}.$$

Next we will consider the weighted modulus of continuity $$\Omega (g;\kappa )$$ defined on $$[0, \infty )$$ as

$$\Omega (g;\kappa )=\sup_{y\geqq 0;\;0< \vert j \vert \leqq \kappa} \frac{ \vert g(y+j)-g(y) \vert }{(1+j^{2})\zeta (y)} \quad \bigl( \forall \; g \in C^{*}_{\zeta}[0, \infty \bigr) ).$$

It is know that for any $$\mu \in [0,\infty )$$, the following inequality:

\begin{aligned} \Omega (g;\mu \kappa )\leqq 2(1+\mu ) \bigl(1+\kappa ^{2}\bigr) \Omega (g;\kappa ) \end{aligned}

holds true $$\forall g\in C^{*}_{\zeta}[0,\infty )$$, and

\begin{aligned} \bigl\vert g(s)-g(y) \bigr\vert \leqq 2 \biggl(\frac{ \vert s-y \vert }{\kappa} +1 \biggr) \bigl(1+\kappa ^{2}\bigr) \Omega (g;\kappa ) \bigl(1+y^{2}\bigr) \bigl(1+(s-y)^{2}\bigr). \end{aligned}

### Theorem 3.1

For $$g \in C^{*}_{\zeta}[0,\infty )$$,

\begin{aligned} \lim_{n\to \infty} \bigl\Vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\Vert _{ \rho}=0. \end{aligned}

### Proof

We will achieve our result from the Korovkin-type theorem and relations

$$\lim_{n} \bigl\Vert G^{(1)}_{n,p,q}e_{i}-e_{i} \bigr\Vert _{\zeta} =0\quad (i=0),$$

which follows from Corollary 1.4.

In what follows, we will prove it for $$i = 1$$ and $$i= 2$$. Letting $$g \in C^{*}_{\zeta}[0,\infty )$$, we get

\begin{aligned} \bigl\Vert G^{(1)}_{n,p_{n},q_{n}}e_{1}-e_{1} \bigr\Vert _{\zeta} =& \sup_{y \geqq 0} \biggl\{ \frac{ \vert G^{(1)}_{n,p_{n},q_{n}}e_{1}-e_{1} \vert }{ \zeta (y)} \biggr\} \\ \leqq &\sup_{y\geqq 0} \frac{ \vert \frac{[n]_{p_{n}q_{n}}(1-p_{n}^{2})-[2]_{p_{n}q_{n}}q_{n}^{n}}{[n+2]_{p_{n}q_{n}}}y \vert }{\zeta (y)} \\ \leq& \sup_{y\geqq 0} \frac{ \vert \frac{[n]_{p_{n}q_{n}}}{[n+2]_{p_{n}q_{n}}} \vert }{\zeta (y)} =0. \end{aligned}

Using a similar consideration, we have

\begin{aligned} \bigl\Vert G^{(1)}_{n,p_{n},q_{n}}e_{2}-e_{2} \bigr\Vert _{\zeta} =& \sup_{y \geqq 0} \biggl\{ \frac{ \vert G^{(1)}_{n,p_{n},q_{n}}e_{2}-e_{2} \vert }{ \zeta (y)} \biggr\} \\ \leqq& \sup_{y\geqq 0} \biggl\{ \frac{ \vert \frac{[n]_{p_{n}q_{n}}([n]_{p_{n}q_{n}} +(p_{n}^{2}-2)[n+1]_{p_{n}q_{n}})+[2]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}q_{n}^{n}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{2} \vert }{\zeta (y)} \biggr\} \\ =& \frac{\frac{ \vert [n]_{p_{n}q_{n}}^{2}-2[n]_{p_{n},q_{n}}[n+1]_{p_{n},q_{n}} +[n+1]_{p_{n},q_{n}}[n+2]_{p_{n},q_{n}} \vert }{[n+1]_{p_{n},q_{n}}[n+2]_{p_{n},q_{n}}}}{\zeta (y)}=0. \end{aligned}

We thus conclude that

$$\lim_{n \to \infty} \bigl\Vert G^{(1)}_{n,p_{n},q_{n}}e_{i}-e_{i} \bigr\Vert _{ \zeta}=0 \quad (i=0,1,2).$$

â€ƒâ–¡

### Theorem 3.2

Let $$g \in C^{*}_{\zeta}[0, \infty )$$. Then,

\begin{aligned} \sup_{y\in [0,\infty )}{ \frac{ \vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \vert }{ (1+y^{2})(1+ {F} y^{4})}} \leqq S \Omega \bigl(g;n^{-\frac{1}{4}} \bigr) \end{aligned}

for large n, where S is a constant, and $$F>0$$ is constants dependent only on n, p, q.

### Proof

For $$y \in [0, \infty )$$, we have

$$G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) = \int _{0}^{\infty}K_{n;p_{n},q_{n}}(y,v)\bigl[g \bigl([n]_{p_{n},q_{n}}v\bigr)-g(y)\bigr]\,d_{p_{n},q_{n}}v.$$

Then,

\begin{aligned} & \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert \\ &\quad \leqq \int _{0}^{\infty} K_{n,p_{n},q_{n}}(y,v) 2\bigl(1+ \kappa _{n}^{2}\bigr) \Omega (g;\kappa _{n}) \bigl(1+y^{2}\bigr) \\ &\qquad {}\times \cdot \biggl( \frac{ \vert [n]_{p_{n},q_{n}}v-y \vert }{\kappa _{n}} +1 \biggr) \bigl(1+ \bigl([n]_{p_{n},q_{n}}v -y \bigr)^{2} \bigr)\,d_{p_{n},q_{n}}v. \end{aligned}

Let us define

$$S(v,p_{n},q_{n},y)= \biggl( \frac{ \vert [n]_{p_{n},q_{n}}v-y \vert }{\kappa _{n}} +1 \biggr) \bigl(1+ \bigl([n]_{p_{n},q_{n}}v -y \bigr)^{2} \bigr).$$

Then,

$$S(v,p_{n},q_{n},y)\leqq \textstyle\begin{cases} 2(1+\kappa _{n}^{2}) & ( \vert 1+([n]_{p_{n},q_{n}}v -y) \vert \leqq \kappa _{n} ), \\ 2(1+\kappa _{n}^{2}) \frac{([n]_{p_{n},q_{n}}v -y)^{4}}{\kappa _{n}^{4}} & ( \vert [n]_{p_{n},q_{n}}v-y \vert \geqq \kappa _{n} ), \end{cases}$$

and

$$S(v,p_{n},q_{n},y)\leqq 2\bigl(1+\kappa _{n}^{2}\bigr) \biggl( 1+ \frac{([n]_{p_{n},q_{n}}v- y)^{4}}{\kappa _{n}^{4}} \biggr).$$

So, clearly, we get

\begin{aligned} & \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert \\ &\quad \leqq 4\bigl(1+\kappa _{n}^{2} \bigr)^{2} \Omega (g;\kappa _{n}) \bigl(1+y^{2} \bigr) \int _{0}^{\infty}K_{n,p_{n},q_{n}}(y,v) \cdot \biggl( 1+\frac{([n]_{p_{n},q_{n}}v -y)^{4}}{\kappa _{n}^{4}} \biggr)\,d_{p_{n},q_{n}}v. \end{aligned}

From Lemma 1.6, it yields

\begin{aligned}& \int _{0}^{\infty} K_{n,p_{n},q_{n}}(y,v) \biggl( 1+ \frac{([n]_{p_{n},q_{n}}v -y)^{4}}{\kappa _{n}^{4}} \biggr)\,d_{p_{n},q_{n}}v \\& \quad = \int _{0}^{\infty} K_{n,p_{n},q_{n}}(y,v) \,d_{p_{n},q_{n}}v + \int _{0}^{\infty} K_{n,p_{n},q_{n}}(y,v) \frac{([n]_{p_{n},q_{n}}v -y)^{4}}{\kappa _{n}^{4}}\,d_{p_{n},q_{n}}v \\& \quad = 1 \\& \qquad {}+ \frac{1}{\kappa _{n}^{4}} \biggl( \frac{[n]_{p_{n}q_{n}}([n]_{p_{n}q_{n}}^{2}+2[n]_{p_{n}q_{n}}[n-1]_{p_{n}q_{n}}+(p_{n}^{2}-4)[n-1]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}})+[2]_{p_{n}q_{n}}[n-1]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}q_{n}^{n}}{[n-1]_{p_{n}q_{n}}[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{4} \biggr). \end{aligned}

For $$\kappa _{n}=n^{-\frac{1}{4}}$$, we get

\begin{aligned} \bigl\vert G^{(1)}_{n,p_{n},q_{n}}(g;y)-g(y) \bigr\vert &\leqq {S}\Omega \bigl(g;n^{-\frac{1}{4}}\bigr) \bigl(1+y^{2}\bigr) \bigl(1+ F y^{4}\bigr). \end{aligned}

â€ƒâ–¡

## 4 Shape-preserving properties

Next we will prove that modified $$(p,q)$$-Gamma-type operators preserve the monotonicity and convexity under certain conditions. We start with

### Theorem 4.1

Let $$g \in C[0,\infty )$$. If $$g^{\prime } (x)>0$$ and g convex on $$[0,\infty )$$, then modified $$(p_{n},q_{n})$$-Gamma-type operators are increasing.

### Proof

We will prove our result in two steps.

Step one. In this case, we will prove the monotonicity of modified $$(p_{n},q_{n})$$-Gamma-type operators for the Lagrange interpolation polynomial of function $$g (y)$$. Let us suppose that $$y_{0}$$, $$y_{1}$$ are distinct numbers in the interval $$[t,z]$$, where $$t< y_{0}< y_{1}< z$$. Then, the Lagrangian interpolation polynomial through points $$(y_{0},g(y_{0}))$$ and $$(y_{1},g (y_{1}))$$ is:

$$P(y)=\frac{y-y_{1}}{y_{0}-y_{1}}g (y_{0})+\frac{y-y_{0}}{y_{1}-y_{0}}g (y_{1}).$$

Based on Corollary 1.4, we have:

$$G^{(1)}_{n,p_{n},q_{n}}{ (P,t)}-G^{(1)}_{n,p_{n},q_{n}}{ (P,z)}=(t-z) \frac{g (y_{0})-g (y_{1})}{y_{0}-y_{1}} \frac{[n]_{p_{n}q_{n}}}{[n+2]_{p_{n}q_{n}}}< 0,$$

which proves that $$G^{(1)}_{n,p_{n},q_{n}}(P(s),y)$$ is also increasing.

Step two. From the above condition, it follows

$$g (y)=P(y)+\frac{g ^{{\prime \prime }}(\xi _{y})}{2!}(y-y_{0}) (y-y_{1}),$$

for number $$\xi _{y}\in (\min_{{}}\{y_{0},y_{1}\},\max_{{}}\{y_{0},y_{1}\})$$. For $$t< y_{0}< y_{1}< z$$ and Corollary 1.4, we have

\begin{aligned}& G^{(1)}_{n,p_{n},q_{n}}{ (g ,t)}-G^{(1)}_{n,p_{n},q_{n}}{ (g ,z)} \\& \quad = \bigl[ G^{(1)}_{n,p_{n},q_{n}}{ (P,t)}-G^{(1)}_{n,p_{n},q_{n}}{ (P,z)} \bigr]\\& \qquad {}+\frac{g ^{{\prime \prime }}(\xi _{s})}{2!} \bigl[ G^{(1)}_{n,p_{n},q_{n}}{ \bigl(s^{2},y\bigr)}-(y_{0}+y_{1})G^{(1)}_{n,p_{n},q_{n}}{ (s,y)}+y_{0}y_{1} \bigr]\\& \qquad {}-\frac{g ^{{\prime \prime }}(\xi _{s})}{2!} \bigl[ G^{(1)}_{n,p_{n},q_{n}}{ \bigl(s^{2},z\bigr)}-(y_{0}+y_{1})G^{(1)}_{n,p_{n},q_{n}}{ (s,z)}+y_{0}y_{1} \bigr] \\& \quad = \bigl[ G^{(1)}_{n,p_{n},q_{n}}{ (P,t)}-G^{(1)}_{n,p_{n},q_{n}}{ (P,z)} \bigr] \\& \qquad {}+(t-z)\frac{[n]_{p_{n}q_{n}}}{[n+1]_{p_{n}q_{n}}} \frac{g ^{{\prime \prime }}(\xi _{s})}{2!} \biggl[ (t+z) \frac{[n]_{p_{n}q_{n}}}{[n+1]_{p_{n}q_{n}}}-(y_{0}+y_{1}) \biggr] < 0. \end{aligned}

Therefore, it proves the theorem.â€ƒâ–¡

### Question

Prove that the above theorem is valid just only if $$f^{\prime } (x)>0$$, on $$[0,\infty )$$.

Thus, the next results show that modified $$(p,q)$$-Gamma-type operators preserve the convexity.

### Theorem 4.2

Let $$g \in C[0,\infty )$$. If $$g (y)$$ is convex on $$[0,\infty )$$, then $$(p_{n},q_{n})$$-Gamma-type operators are also convex.

### Proof

Let us consider that $$g ^{{\prime \prime}}(y)>0$$. Then,

$$\bigl[ G^{(1)}_{n,p_{n},q_{n}}{ \bigl(P(s),y\bigr)} \bigr]_{y} ^{{\prime \prime }}= \biggl[\frac{g(y_{0})-g(y_{1})}{y_{0}-y_{1}} \frac{[n]_{p_{n}q_{n}}}{[n+1]_{p_{n}q_{n}}}y- \frac{y_{1}g(y_{0})}{y_{0}-y_{1}}-\frac{y_{0}g(y_{1})}{y_{1}-y_{0}} \biggr]^{{\prime \prime }}=0.$$

On the other hand,

\begin{aligned}& G^{(1)}_{n,p_{n},q_{n}}{ \bigl(g(s) ,y\bigr)} \\& \quad =G^{(1)}_{n,p_{n},q_{n}}{ \bigl(P(s),y\bigr)}+ \frac{g ^{{\prime \prime }}(\xi _{s})}{2!} \bigl[ G^{(1)}_{n,p_{n},q_{n}}{ \bigl(s^{2},y\bigr)}-(y_{0}+y_{1})G^{(1)}_{n,p_{n},q_{n}}{ (s,y)}+y_{0}y_{1} \bigr]\\& \quad =G^{(1)}_{n,p_{n},q_{n}}{ (P,y)}+ \frac{g ^{{\prime \prime }}(\xi _{s})}{2!} \biggl[ \frac{[n]_{p_{n}q_{n}}^{2}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}}y^{2}-(y_{0}+y_{1}) \frac{[n]_{p_{n}q_{n}}}{[n+2]_{p_{n}q_{n}}}y+y_{0}y_{1} \biggr]. \end{aligned}

From the last relation, it follows

$$\bigl[ G^{(1)}_{n,p_{n},q_{n}}{ \bigl(g(s) ,y\bigr)} \bigr]_{y} ^{{\prime \prime }}=g ^{{\prime \prime }}(\xi _{s})\cdot \frac{[n]_{p_{n}q_{n}}^{2}}{[n+1]_{p_{n}q_{n}}[n+2]_{p_{n}q_{n}}} >0.$$

Hence, it proves the theorem.â€ƒâ–¡

Not applicable.

## References

1. Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Application. Walter de Gruyter Studies in Math., vol.Â 17. de Gruyter & Co., Berlin (1994)

2. Atlihan, O.G., Unver, M., Duman, O.: Korovkin theorems on weighted spaces: revisited. Period. Math. Hung. 75(2), 201â€“209 (2017)

3. Braha, N.L.: Some weighted equi-statistical convergence and Korovkin type-theorem. Results Math. 70(3â€“4), 433â€“446 (2016)

4. Braha, N.L.: Some properties of new modified SzÃ¡sz-Mirakyan operators in polynomial weight spaces via power summability method. Bull. Math. Anal. Appl. 10(3), 53â€“65 (2018)

5. Braha, N.L.: Some properties of Baskakov-Schurer-SzÃ¡sz operators via power summability methods. Quaest. Math. 42, 1411â€“1426 (2019)

6. Braha, N.L.: Korovkin type theorem for Bernstein-Kantorovich operators via power summability method. Anal. Math. Phys. 10(4), 62 (2020)

7. Braha, N.L.: Some properties of modified SzÃ¡sz-Mirakyan operators in polynomial spaces via the power summability method. J. Appl. Anal. 26(1), 79â€“90 (2020)

8. Braha, N.L., Kadak, U.: Approximation properties of the generalized Szasz operators by multiple Appell polynomials via power summability method. Math. Methods Appl. Sci. 43(5), 2337â€“2356 (2020)

9. Braha, N.L., Loku, V.: Korovkin type theorems and its applications via Î±Î²-statistically convergence. J. Math. Inequal. 14(4), 951â€“966 (2020)

10. Braha, N.L., Loku, V.: Korovkin type theorems and its applications via Î±Î²-statistically convergence. J. Math. Inequal. 14(4), 951â€“966 (2020)

11. Bukweli-Kyemba, J.D., Hounkonnou, M.N.: Quantum deformed algebras: coherent states and special functions (2013). arXiv:1301.0116v1

12. Campiti, M., Metafune, G.: $$L^{p}$$-convergence of Bernstein-Kantorovich-type operators. Ann. Pol. Math. 63(3), 273â€“280 (1996)

13. Corcino, R.B.: On p, q-binomial coefficients. Integers 8, A29 (2008)

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

15. Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer, New York (1987)

16. Duman, O., Khan, M.K., Orhan, C.: A-Statistical convergence of approximating operators. Math. Inequal. Appl. 4, 689â€“699 (2003)

17. Gavrea, I., RaÅŸa, I.: Remarks on some quantitative Korovkin-type results. Rev. Anal. NumÃ©r. ThÃ©or. Approx. 22, 173â€“176 (1993)

18. Ispir, N.: On modified Baskakov operators on weighted spaces. Turk. J. Math. 25, 355â€“365 (2001)

19. Jagannathan, R., Rao, K.S.: Tow-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series (2006). arXiv:math/0602613v

20. Kadak, U., Braha, N.L., Srivastava, H.M.: Statistical weighted $$\mathcal {B}$$-summability and its applications to approximation theorems. Appl. Math. Comput. 302, 80â€“96 (2017)

21. Kumar Singh, J., Narain Agrawal, P., Kajla, A.: Approximation by modified q-Gamma type operators via A-statistical convergence and power series method. Linear Multilinear Algebra, 1â€“20 (2021)

22. Landau, E.: Einige Ungleichungen fÃ¼r zweÃ­mal differentzierban funktionen. Proc. Lond. Math. Soc. 13, 43â€“49 (1913)

23. Loku, V., Braha, N.L.: Some weighted statistical convergence and Korovkin type-theorem. J. Inequal. Spec. Funct. 8(3), 139â€“150 (2017)

24. Loku, V., Braha, N.L., Mansour, T., Mursaleen, M.: Approximation by a power series summability method of Kantorovich type SzÃ¡sz operators including Sheffer polynomials. Adv. Differ. Equ. 2021, 165 (2021)

25. Lupas, A., Muller, M.: Approximations eigenschaften der gamma operatoren. Math. Z. 98, 208â€“226 (1967)

26. Njionou Sadjang, P.: On the $$(p, q)$$-Gamma and the $$(p, q)$$-Beta functions (2015). arXiv:1506.07394v1 [math.NT] 22 Jun 2015

27. Ã–zarslan, M.A., AktuÄŸlu, H.: Local approximation for certain King type operators. Filomat 27(1), 173â€“181 (2013)

28. Peetre, J.: Theory of interpolation of normed spaces. Notas Mat. Rio de Janeiro 39, 1â€“86 (1963)

29. Usta, F., Betus, O.: A new modification of gamma operators with a better error estimation. Linear Multilinear Algebra (2020). https://doi.org/10.1080/03081087.2020.1791033

30. Zhuk, V.V.: Functions of the $${\mathrm{Lip}}\,1$$ class and S. N. Bernsteinâ€™s polynomials (Russian), Vestn. Leningr. Univ., Math. Mekh. Astronom. 1989(1) (1989), 25â€“30, 122â€“123; Vestn. Leningr. Univ., Math. 22 (1989), 38â€“44

Not applicable.

## Author information

Authors

### Contributions

All authors has contributede equaly

### Corresponding author

Correspondence to Naim Latif Braha.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisherâ€™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 licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the articleâ€™s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articleâ€™s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

Braha, N.L. Approximation by modified $$(p,q)$$-gamma-type operators. J Inequal Appl 2024, 37 (2024). https://doi.org/10.1186/s13660-024-03109-1