Convergence of the q-Stancu-Szász-Beta type operators

Dinlemez, Ülkü

doi:10.1186/1029-242X-2014-354

Convergence of the q-Stancu-Szász-Beta type operators

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Published: 24 September 2014

Volume 2014, article number 354, (2014)
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Convergence of the q-Stancu-Szász-Beta type operators

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Ülkü Dinlemez¹

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Abstract

In this paper, we study on q-Stancu-Szász-Beta type operators. We give these operators convergence properties and obtain a weighted approximation theorem in the interval $[0, \infty)$ .

MSC:41A25, 41A36.

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Approximation of q-Stancu-Beta Operators Which Preserve $$x^2$$

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q-Szász–Durrmeyer Type Operators Based on Dunkl Analogue

Article 28 June 2018

1 Introduction

In [1], Mahmudov constructed q-Szász operators and obtained rate of global convergence in the frame of weighted spaces and a Voronovskaja type theorem for these operators. In [2], Gupta and Mahmudov studied on the q-analog of the Szász-Beta type operators. In [3], Yüksel and Dinlemez gave a Voronovskaja type theorem for q-analog of a certain family Szász-Beta type operators. In [4], Govil and Gupta introduced the q-analog of certain Beta-Szász-Stancu operators. They estimated the moments and established direct results in terms of modulus of continuity and an asymptotic formula for the q-operators. In [5–14], interesting generalization about q-calculus were given. Our aims are to give approximation properties and a weighted approximation theorem for q-Stancu-Szász-Beta type operators. We use without further explanation the basic notations and formulas, from the theory of q-calculus as set out in [15–19]. Let $A > 0$ and f be a real valued continuous function defined on the interval $[0, \infty)$ . For $0 < q \leq 1$ , q-Stancu-Szász-Beta type operators are defined as

B_{n, q}^{(α, β)} (f, x) = \sum_{k = 0}^{\infty} s_{n, k}^{q} (x) \int_{0}^{\infty / A} b_{n, k}^{q} (t) f (\frac{{[n]}_{q} t + α}{{[n]}_{q} + β}) d_{q} t,

(1.1)

where

s_{n, k}^{q} (x) = {({[n]}_{q} x)}^{k} \frac{e^{- {[n]}_{q} x}}{{[k]}_{q}!}

and

b_{n, k}^{q} (x) = \frac{q^{k^{2}} x^{k}}{B_{q} (k + 1, n) {(1 + x)}_{q}^{n + k + 1}} .

If we write $q = 1$ and $α = β = 0$ in (1.1), then the operators $B_{n, q}^{(α, β)} (f, x)$ are reduced to Szász-Beta type operators studied in [20–23].

2 Auxiliary results

For the sake of brevity, the notation $F_{s}^{q} (n) = \prod_{i = 1}^{s} {[n - i]}_{q}$ and $G_{β}^{q} (n) = ({[n]}_{q} + β)$ will be used throughout the article. Now we are ready to give the following lemma for the Korovkin test functions.

Lemma 1 Let $e_{m} (t) = t^{m}$ , $m = 0, 1, 2$ , we get

\begin{array}{rcl} (i) & B_{n, q}^{(α, β)} (e_{0}, x) = 1, \\ (ii) & B_{n, q}^{(α, β)} (e_{1}, x) = \frac{{[n]}_{q}^{2} x}{q^{2} G_{β}^{q} (n) F_{1}^{q} (n)} + \frac{{[n]}_{q}}{q G_{β}^{q} (n) F_{1}^{q} (n)} + \frac{α}{G_{β}^{q} (n)}, \\ (iii) & B_{n, q}^{(α, β)} (e_{2}, x) = \frac{{[n]}_{q}^{4} x^{2}}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + {\frac{{[n]}_{q}^{3}}{q^{5} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} \\ + \frac{(1 + {[2]}_{q}) {[n]}_{q}^{3}}{q^{4} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + \frac{2 α {[n]}_{q}^{2}}{q^{2} G_{β}^{q} {(n)}^{2} F_{1}^{q} (n)}} x \\ + \frac{{[2]}_{q} {[n]}_{q}^{2}}{q^{3} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + \frac{2 α {[n]}_{q}}{q G_{β}^{q} {(n)}^{2} F_{1}^{q} (n)} + \frac{α^{2}}{G_{β}^{q} {(n)}^{2}} . \end{array}

Proof Using the q-Gamma and q-Beta functions in [15, 24], we obtain the following equality:

\begin{matrix} q^{k^{2}} \int_{0}^{\infty / A} \frac{1}{B (k + 1, n)} \frac{t^{k + m}}{{(1 + t)}_{q}^{n + k + 1}} d_{q} t \\ = \frac{{[m + k]}_{q}! {[n - m - 1]}_{q}! q^{{2 k^{2} - (k + m) (k + m + 1)} / 2}}{{[k]}_{q}! {[n - 1]}_{q}!} . \end{matrix}

(2.1)

Then, using (2.1), for $m = 0$ , we get

\begin{array}{rcl} B_{n, q}^{(α, β)} (e_{0}, x) & = & e^{- {[n]}_{q} x} \sum_{k = 0}^{\infty} \frac{{({[n]}_{q} x)}^{k}}{{[k]}_{q}!} q^{k (k - 1) / 2} \\ = & e^{- {[n]}_{q} x} E_{q}^{{[n]}_{q} x} = 1, \end{array}

and the proof of (i) is finished. With a direct computation, we obtain (ii) as follows:

\begin{array}{rcl} B_{n, q}^{(α, β)} (e_{1}, x) & = & \frac{{[n]}_{q}}{G_{β}^{q} (n) F_{1}^{q} (n)} \sum_{k = 1}^{\infty} \frac{{({[n]}_{q} x)}^{k}}{{[k - 1]}_{q}!} q^{k (k - 3) - 2 / 2} e^{- {[n]}_{q} x} \\ + \frac{{[n]}_{q}}{G_{β}^{q} (n) F_{1}^{q} (n)} \sum_{k = 0}^{\infty} \frac{{({[n]}_{q} x)}^{k}}{{[k]}_{q}!} q^{k (k - 1) - 2 / 2} e^{- {[n]}_{q} x} \\ + \frac{α}{G_{β}^{q} (n)} \sum_{k = 0}^{\infty} \frac{{({[n]}_{q} x)}^{k}}{{[k]}_{q}!} q^{k (k - 1) / 2} e^{- {[n]}_{q} x} \\ = & \frac{{[n]}_{q}^{2} x}{q^{2} G_{β}^{q} (n) F_{1}^{q} (n)} E_{q}^{{[n]}_{q} x} e^{- {[n]}_{q} x} + \frac{{[n]}_{q}}{q G_{β}^{q} (n) F_{1}^{q} (n)} E_{q}^{{[n]}_{q} x} e^{- {[n]}_{q} x} \\ + \frac{α}{G_{β}^{q} (n)} E_{q}^{{[n]}_{q} x} e^{- {[n]}_{q} x} \\ = & \frac{{[n]}_{q}^{2} x}{q^{2} G_{β}^{q} (n) F_{1}^{q} (n)} + \frac{{[n]}_{q}}{q G_{β}^{q} (n) F_{1}^{q} (n)} + \frac{α}{G_{β}^{q} (n)} . \end{array}

Using the equality

{[n]}_{q} = {[s]}_{q} + q^{s} {[n - s]}_{q}, 0 \leq s \leq n,

(2.2)

we get

\begin{array}{rcl} B_{n, q}^{(α, β)} (e_{2}, x) & = & \frac{{[n]}_{q}^{4} x^{2}}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} \\ + {\frac{{[n]}_{q}^{3}}{q^{5} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + \frac{(1 + {[2]}_{q}) {[n]}_{q}^{3}}{q^{4} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + \frac{2 α {[n]}_{q}^{2}}{q^{2} G_{β}^{q} {(n)}^{2} F_{1}^{q} (n)}} x \\ + \frac{{[2]}_{q} {[n]}_{q}^{2}}{q^{3} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + \frac{2 α {[n]}_{q}}{q G_{β}^{q} {(n)}^{2} F_{1}^{q} (n)} + \frac{α^{2}}{G_{β}^{q} {(n)}^{2}}, \end{array}

and so we have the proof of (iii). □

To obtain our main results we need to compute the second moment.

Lemma 2 Let $q \in (0, 1)$ and $n > 2$ . Then we have the following inequality:

B_{n, q}^{(α, β)} ({(t - x)}^{2}, x) \leq (\frac{2 (1 - q^{4})}{q^{6}} + \frac{164 {(α + β + 1)}^{2} {[n]}_{q}}{q^{6} F_{2}^{q} (n)}) x (x + 1) + \frac{6 {(α + 1)}^{2}}{q^{3} G_{β}^{q} (n)} .

Proof From the linearity of the $B_{n, q}^{(α, β)}$ operators and Lemma 1, we write the second moment as

\begin{matrix} B_{n, q}^{(α, β)} ({(t - x)}^{2}, x) \\ = {\frac{{[n]}_{q}^{4}}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} - \frac{2 {[n]}_{q}^{2}}{q^{2} G_{β}^{q} (n) F_{1}^{q} (n)} + 1} x^{2} \\ + {\frac{{1 + (1 + {[2]}_{q}) q} {[n]}_{q}^{3}}{q^{5} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + \frac{2 α {[n]}_{q}^{2}}{q^{2} G_{β}^{q} {(n)}^{2} F_{1}^{q} (n)} - \frac{2 {[n]}_{q}}{q G_{β}^{q} (n) F_{1}^{q} (n)} - \frac{2 α}{G_{β}^{q} (n)}} x \\ + \frac{{[2]}_{q} {[n]}_{q}^{2}}{q^{3} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + \frac{2 α {[n]}_{q}}{q G_{β}^{q} {(n)}^{2} F_{1}^{q} (n)} + \frac{α^{2}}{G_{β}^{q} {(n)}^{2}} \\ \leq {\frac{{[n]}_{q}^{4}}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} - \frac{2 {[n]}_{q}^{2}}{q^{2} G_{β}^{q} (n) F_{1}^{q} (n)} + 1 + \frac{{1 + (1 + {[2]}_{q}) q} {[n]}_{q}^{3}}{q^{5} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} \\ + \frac{2 α {[n]}_{q}^{2}}{q^{2} G_{β}^{q} {(n)}^{2} F_{1}^{q} (n)}} x (x + 1) + \frac{{[2]}_{q} {[n]}_{q}^{2}}{q^{3} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + \frac{2 α {[n]}_{q}}{q G_{β}^{q} {(n)}^{2} F_{1}^{q} (n)} + \frac{α^{2}}{G_{β}^{q} {(n)}^{2}} \\ \leq {\frac{{[n]}_{q}^{4} (1 + q^{6}) - 2 q^{4} {[n - 2]}_{q}^{4} + 2 β q^{6} {[n]}_{q} {[n - 1]}_{q} {[n - 2]}_{q}}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} \\ + \frac{(q + q^{2} + {[2]}_{q} q^{2}) {[n]}_{q}^{3}}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + \frac{q^{6} β^{2} {[n - 1]}_{q} {[n - 2]}_{q}}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} + \frac{2 α q^{4} {[n]}_{q}^{2} {[n - 2]}_{q}}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)}} x (x + 1) \\ + \frac{{{[2]}_{q} + 2 α q^{2} + α^{2} q^{3}} {[n]}_{q}}{q^{3} G_{β}^{q} (n) F_{2}^{q} (n)} . \end{matrix}

From (2.2), we have

\begin{matrix} B_{n, q}^{(α, β)} ({(t - x)}^{2}, x) \\ \leq {\frac{{[n - 2]}_{q}^{4} (q^{14} + q^{8} - 2 q^{4})}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} \\ + \frac{(1 + q^{6}) {4 {[2]}_{q} q^{6} {[n - 2]}_{q}^{3} + 6 {[2]}_{q}^{2} q^{4} {[n - 2]}_{q}^{2} + 4 {[2]}_{q}^{3} q^{2} {[n - 2]}_{q} + {[2]}_{q}^{4}}}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)} \\ + \frac{(q + q^{2} + {[2]}_{q} q^{2} + 2 β q^{6} + 2 α q^{4}) {[n]}_{q}^{3} + β^{2} q^{6} {[n]}_{q}^{2}}{q^{6} G_{β}^{q} {(n)}^{2} F_{2}^{q} (n)}} x (x + 1) \\ + \frac{({[2]}_{q} + q^{2}) ({[2]}_{q} + 2 α q^{2} + α^{2} q^{3})}{q^{3} G_{β}^{q} (n) F_{1}^{q} (n)} \\ \leq (\frac{2 (1 - q^{4})}{q^{6}} + \frac{164 {(α + β + 1)}^{2} {[n]}_{q}}{q^{6} F_{2}^{q} (n)}) x (x + 1) + \frac{6 {(α + 1)}^{2}}{q^{3} G_{β}^{q} (n)} . \end{matrix}

And the proof of Lemma 2 is now finished. □

3 Direct estimates

Now in our considerations, $C_{B} [0, \infty)$ denotes the set of all bounded-continuous functions from $[0, \infty)$ to ℝ. $C_{B} [0, \infty)$ is a normed space with the norm ${∥ f ∥}_{B} = sup {| f (x) | : x \in [0, \infty)}$ . We denote the first modulus of continuity on the finite interval $[0, b]$ , $b > 0$ ,

ω_{[0, b]} (f, δ) = sup_{0 < h \leq δ, x \in [0, b]} | f (x + h) - f (x) | .

(3.1)

The Peetre K-functional is defined by

K_{2} (f, δ) = inf {{∥ f - g ∥}_{B} + δ {∥ g^{″} ∥}_{B} : g \in W_{\infty}^{2}}, δ > 0,

where $W_{\infty}^{2} = {g \in C_{B} [0, \infty) : g^{'}, g^{″} \in C_{B} [0, \infty)}$ . By Theorem 2.4 in [25], p.177, there exists a positive constant C such that

K_{2} (f, δ) \leq C ω_{2} (f, \sqrt{δ}),

(3.2)

where

ω_{2} (f, \sqrt{δ}) = sup_{0 < h \leq \sqrt{δ}} sup_{x \in [0, \infty)} | f (x + 2 h) - 2 f (x + h) - f (x) | .

Gadzhiev proved the weighted Korovkin-type theorems in [26]. We give the Gadzhiev results in weighted spaces. Let $ρ (x) = 1 + x^{2}$ and the weighted spaces $C_{ρ} [0, \infty)$ denote the space of all continuous functions f, satisfying $| f (x) | \leq M_{f} ρ (x)$ , where $M_{f}$ is a constant depending only on f. $C_{ρ} [0, \infty)$ is a normed space with the norm ${∥ f ∥}_{ρ} = sup {\frac{| f (x) |}{ρ (x)} : x \in R^{+} \cup {0}}$ and $C_{ρ}^{*} [0, \infty)$ denotes the subspace of all functions $f \in C_{ρ} [0, \infty)$ for which ${lim}_{| x | \to \infty} \frac{| f (x) |}{ρ (x)}$ exists finitely.

Thus we are ready to give direct results. The following lemma is routine and its proof is omitted.

Lemma 3 Let

{\bar{B}}_{n, q}^{(α, β)} (f, x) = B_{n, q}^{(α, β)} (f, x) - f (D_{n, q}^{(α, β)} (x)) + f (x) .

(3.3)

Then the following assertions hold for the operators (3.3):

\begin{array}{rcl} (i) & {\bar{B}}_{n, q}^{(α, β)} (1, x) = 1, \\ (ii) & {\bar{B}}_{n, q}^{(α, β)} (t, x) = x, \\ (iii) & {\bar{B}}_{n, q}^{(α, β)} (t - x, x) = 0, \end{array}

where $D_{n, q}^{(α, β)} (x) = \frac{{[n]}_{q}^{2} x}{q^{2} G_{β}^{q} (n) F_{1}^{q} (n)} + \frac{{[n]}_{q}}{q G_{β}^{q} (n) F_{1}^{q} (n)} + \frac{α}{G_{β}^{q} (n)}$ .

Lemma 4 Let $q \in (0, 1)$ and $n > 2$ . Then for every $x \in [0, \infty)$ and $f^{″} \in C_{B} [0, \infty)$ , we have the inequality

| {\bar{B}}_{n, q}^{(α, β)} (f, x) - f (x) | \leq δ_{n, q}^{(α, β)} (x) {∥ f^{″} ∥}_{B},

where $δ_{n, q}^{(α, β)} (x) = (\frac{2 (1 - q^{4})}{q^{6}} + \frac{263 {(α + β + 1)}^{2}}{q^{6} F_{1}^{q} (n)}) x (x + 1) + \frac{5 {(α + 1)}^{2}}{q^{3} G_{β}^{q} (n)}$ .

Proof Using Taylor’s expansion

f (t) = f (x) + (t - x) f^{'} (x) + \int_{x}^{t} (t - u) f^{″} (u) d u

and Lemma 3, we obtain

{\bar{B}}_{n, q}^{(α, β)} (f, x) - f (x) = {\bar{B}}_{n, q}^{(α, β)} (\int_{x}^{t} (t - u) f^{″} (u) d u, x) .

Then, using Lemma 1 and the inequality

| \int_{x}^{t} (t - u) f^{″} (u) d u | \leq {∥ f^{″} ∥}_{B} \frac{{(t - x)}^{2}}{2},

we get

\begin{matrix} | {\bar{B}}_{n, q}^{(α, β)} (f, x) - f (x) | \\ \leq | B_{n, q}^{(α, β)} (\int_{x}^{t} (t - u) f^{″} (u) d u, x) - \int_{x}^{D_{n, q}^{(α, β)} (x)} {D_{n, q}^{(α, β)} (x) - u} f^{″} (u) d u | \\ \leq \frac{{∥ f^{″} ∥}_{B}}{2} {(\frac{2 (1 - q^{4})}{q^{6}} + \frac{164 {(α + β + 1)}^{2} {[n]}_{q}}{q^{6} F_{2}^{q} (n)} + {(\frac{{[n]}_{q}^{2}}{q^{2} G_{β}^{q} (n) F_{1}^{q} (n)} - 1)}^{2} \\ + \frac{2 {[n]}_{q}^{3}}{q^{3} G_{β}^{q} {(n)}^{2} F_{1}^{q} {(n)}^{2}} + \frac{2 {[n]}_{q}^{2} α}{q^{2} G_{β}^{q} {(n)}^{2} F_{1}^{q} (n)}) x (x + 1) + {(\frac{{[n]}_{q} + α q {[n - 1]}_{q}}{q G_{β}^{q} (n) F_{1}^{q} (n)})}^{2} \\ + \frac{6 {(α + 1)}^{2}}{q^{3} G_{β}^{q} (n) F_{1}^{q} (n)}} \\ \leq \frac{{∥ f^{″} ∥}_{B}}{2} {(\frac{4 (1 - q^{4})}{q^{6}} + \frac{526 {(α + β + 1)}^{2}}{q^{6} F_{1}^{q} (n)}) x (x + 1) + \frac{10 {(α + 1)}^{2}}{q^{3} G_{β}^{q} (n)}} . \end{matrix}

And the proof of the Lemma 4 is now completed. □

Theorem 1 Let $(q_{n}) \subset (0, 1)$ a sequence such that $q_{n} \to 1$ as $n \to \infty$ . Then for every $n > 2$ , $x \in [0, \infty)$ and $f \in C_{B} [0, \infty)$ , we have the inequality

| B_{n, q_{n}}^{(α, β)} (f, x) - f (x) | \leq 2 M ω_{2} (f, \sqrt{δ_{n, q_{n}}^{(α, β)} (x)}) + w (f, η_{n, q_{n}}^{(α, β)} (x)),

where $η_{n, q_{n}}^{(α, β)} (x) = (\frac{{[n]}_{q_{n}}^{2}}{q_{n}^{2} G_{β}^{q_{n}} (n) F_{1}^{q_{n}} (n)} - 1) x + \frac{{[n]}_{q_{n}}}{q_{n} G_{β}^{q_{n}} (n) F_{1}^{q_{n}} (n)} + \frac{α}{G_{β}^{q_{n}} (n)}$ .

Proof Using (3.3) for any $g \in W_{\infty}^{2}$ , we obtain the following inequality:

\begin{array}{rcl} | B_{n, q_{n}}^{(α, β)} (f, x) - f (x) | & \leq & | {\bar{B}}_{n, q_{n}}^{(α, β)} (f - g, x) - (f - g) (x) + {\bar{B}}_{n, q_{n}}^{(α, β)} (g, x) - g (x) | \\ + | f (\frac{{[n]}_{q_{n}}^{2}}{q_{n}^{2} G_{β}^{q_{n}} (n) F_{1}^{q_{n}} (n)} x + \frac{{[n]}_{q_{n}}}{q_{n} G_{β}^{q_{n}} (n) F_{1}^{q_{n}} (n)} + \frac{α}{G_{β}^{q_{n}} (n)}) - f (x) | . \end{array}

From Lemma 4, we get

\begin{array}{rcl} | B_{n, q_{n}}^{(α, β)} (f, x) - f (x) | & \leq & 2 {∥ f - g ∥}_{B} + δ_{n, q_{n}}^{(α, β)} (x) ∥ g^{″} ∥ \\ + | f (\frac{{[n]}_{q_{n}}^{2}}{q_{n}^{2} G_{β}^{q_{n}} (n) F_{1}^{q_{n}} (n)} x + \frac{{[n]}_{q_{n}}}{q_{n} G_{β}^{q_{n}} (n) F_{1}^{q_{n}} (n)} + \frac{α}{G_{β}^{q_{n}} (n)}) - f (x) | . \end{array}

By using equality (3.1) we have

| B_{n, q_{n}}^{(α, β)} (f, x) - f (x) | \leq 2 {∥ f - g ∥}_{B} + δ_{n, q_{n}}^{(α, β)} (x) {∥ g^{″} ∥}_{B} + w (f, η_{n, q_{n}}^{(α, β)} (x)) .

Taking the infimum over $g \in W_{\infty}^{2}$ on the right-hand side of the above inequality and using the inequality (3.2), we get the desired result. □

Theorem 2 Let $(q_{n}) \subset (0, 1)$ a sequence such that $q_{n} \to 1$ as $n \to \infty$ . Then $f \in C_{ρ}^{*} [0, \infty)$ , and we have

lim_{n \to \infty} {∥ B_{n, q_{n}}^{(α, β)} (f) - f ∥}_{ρ} = 0 .

Proof From Lemma 1, it is obvious that ${∥ B_{n, q_{n}}^{(α, β)} (e_{0}) - e_{0} ∥}_{ρ} = 0$ . Since $| \frac{{[n]}_{q_{n}}^{2}}{q_{n}^{2} G_{β}^{q_{n}} (n) F_{1}^{q_{n}} (n)} x + \frac{{[n]}_{q_{n}}}{q_{n} G_{β}^{q_{n}} (n) F_{1}^{q_{n}} (n)} + \frac{α}{G_{β}^{q_{n}} (n)} - x | \leq (x + 1) o (1)$ and $\frac{x + 1}{1 + x^{2}}$ is positive and bounded from above for each $x \geq 0$ , we obtain

{∥ B_{n, q_{n}}^{(α, β)} (e_{1}) - e_{1} ∥}_{ρ} \leq \frac{x + 1}{1 + x^{2}} o (1) .

And then ${lim}_{n \to \infty} {∥ B_{n, q_{n}}^{(α, β)} (e_{1}) - e_{1} ∥}_{ρ} = 0$ .

Similarly for every $n > 2$ , we write

\begin{array}{rcl} {∥ B_{n, q_{n}}^{(α, β)} (e_{2}) - e_{2} ∥}_{ρ} & = & sup_{x \in [0, \infty)} {\frac{| (\frac{{[n]}_{q_{n}}^{4}}{q_{n}^{6} G_{β}^{q_{n}} {(n)}^{2} F_{2}^{q_{n}} (n)} - 1) x^{2}}{1 + x^{2}} \\ + \frac{{\frac{(1 + (1 + {[2]}_{q_{n}}) q_{n}) {[n]}_{q_{n}}^{3} + 2 α q^{2} {[n]}_{q_{n}}^{2} {[n - 1]}_{q_{n}}}{q_{n}^{5} G_{β}^{q_{n}} {(n)}^{2} F_{2}^{q_{n}} (n)}} x + \frac{{[2]}_{q_{n}} {[n]}_{q_{n}}^{2}}{q_{n}^{3} G_{β}^{q_{n}} {(n)}^{2} F_{2}^{q_{n}} (n)}}{1 + x^{2}} \\ + \frac{\frac{+ 2 α q_{n}^{2} {[n]}_{q_{n}} {[n - 2]}_{q_{n}}}{q_{n}^{3} G_{β}^{q_{n}} {(n)}^{2} F_{2}^{q_{n}} (n)} + \frac{α^{2}}{G_{β}^{q_{n}} {(n)}^{2}} |}{1 + x^{2}}} \\ \leq & sup_{x \in [0, \infty)} \frac{1 + x + x^{2}}{1 + x^{2}} o (1), \end{array}

we get ${lim}_{n \to \infty} {∥ B_{n, q_{n}}^{(α, β)} (e_{2}) - e_{2} ∥}_{ρ} = 0$ . Thus, from AD Gadzhiev’s theorem in [26], we obtain the desired result of Theorem 2. □

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Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, Ankara, 06500, Turkey
Ülkü Dinlemez

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Ülkü Dinlemez
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Dinlemez, Ü. Convergence of the q-Stancu-Szász-Beta type operators. J Inequal Appl 2014, 354 (2014). https://doi.org/10.1186/1029-242X-2014-354

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Received: 02 April 2014
Accepted: 27 August 2014
Published: 24 September 2014
DOI: https://doi.org/10.1186/1029-242X-2014-354

Convergence of the q-Stancu-Szász-Beta type operators

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Convergence of the q-Stancu-Szász-Beta type operators

Abstract

Similar content being viewed by others

Approximation by Kantorovich Type q-Bernstein-Stancu Operators

Approximation of q-Stancu-Beta Operators Which Preserve $$x^2$$

q-Szász–Durrmeyer Type Operators Based on Dunkl Analogue

1 Introduction

2 Auxiliary results

3 Direct estimates

References

Acknowledgements

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Corresponding author

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