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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 [48, 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. 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. □

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. □

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

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

 □

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)

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

 □

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, [29, 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]. □

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 Cauchy–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. □

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

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

 □

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. □

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. □

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). □

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

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

 □

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. □

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 Cauchy–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 Hölder 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 Cauchy–Schwarz inequality, we have:

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

 □

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

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

 □

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

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

 □

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. □

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. □

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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

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