PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 211–220 DOI: https://doi.org/10.2298/PIM1716211A

APPLICATIONS OF (p, q) -GAMMA FUNCTION TO SZÁSZ DURRMEYER OPERATORS

Ali Aral and Vijay Gupta

Abstract. We define a (p, q) analogue of Gamma function. As an application, we propose (p, q)-Szász–Durrmeyer operators, estimate moments and establish some direct results.

1. Introduction

In the last two decades the quantum calculus is an active area of research among researchers. The quantum calculus find applications in a number of areas, including approximation theory. The relationship between approximation theory and q-calculus encouraged the mathematicians to give q-analogue of known results (see [3]). This rapid development of q-calculus has led to the discovery of new generalization of this theory. This produces some advantages like that the rate of convergence of q-operators is more flexible and better than the classical one. Since the q-calculus is based on one parameter, there is a possibility of extension of q-calculus. In this direction Sahai–Yadav [14] established some extensions to post-quantum calculus in special functions. A question arises: can we modify the operators using (p, q)-calculus such that our modified operator has better error estimation than the classical ones. For this purpose, we will define (p, q)-Szász– Durrmeyer operators. Several well-known operators may extend to (p, q)-analogues. Mursaleen et al introduced the (p, q)-analogue of the Bernstein operators in [10]. There both point-wise convergence and asymptotic formula are considered. Other important class of discrete operators has been investigated by using (p, q)-calculus. For example (p, q)-Bernstein–Stancu operators appeared in [9] (p, q) Bleimann– Butzer–Hahn and (p, q)-Szász Mirakyan operators have been studied recently in [1, 11]. Very recently, in order to obtain an approximation process in the space of (p, q)-Bernstein operators, the authors [5] defined Durrmeyer type modification of (p, q)-Bernstein operators.

2010 Mathematics Subject Classification: Primary 33B15; Secondary 41A25.

Key words and phrases: (p, q)-Gamma function, (p, q)-Szász–Durrmeyer operators, direct estimates, modulus of continuity.

Communicated by Gradimir Milovanović.

Motivated by all the above results we propose Durrmeyer type modification of the (p, q)-Szász Mirakyan operators using an integral version of (p, q)-Gamma function (as we know it is first in literature).

The paper is organized as follows: the next section contains some basic facts regarding(p, q)-calculus, we also introduce (p, q)-analogue of Gamma function. The construction of the announced class of operators is presented in Section 3. Section 4 deals with the quantitative type estimate with a suitable modulus of continuity. The last section is devoted to weighted Korovin type theorems and we estimate the approximation of bounded functions by announced operators with the help of a Lipschitz-type maximal function.

2. Notations and Preliminaries

Following the definitions and notations of [14]: Set ℕ_{0} = {0} ∪ ℕ, the (p, q)-numbers are defined as

[n]_{p,q} = p^{n−1} + p^{n−2} q + p^{n−3} q^{2} + ⋯+ pq^{n−2} + q^{n−1} = \frac{p^{n} − q^{n}}{p − q}

for n ∈ ℕ. The (p, q)-factorial [n]_{p,q}! of the element n ∈ ℕ means

[n]_{p,q}! = ∏_{k=1}^{n} [k]_{p,q}, n ⩾ 1, [0]_{p,q}! = 1.

The (p, q)-binomial theorem is given by

1Φ_{0}((a, b); −; (p, q), x) = \frac{((p, bx); −; (p, q))_{∞}}{((p, ax); −; (p, q))_{∞}} ,

where ((a, b); −; (p, q))_{∞} = ∏ _{n=0}^{∞} (ap^{n}−bq^{n}). Two different (p, q)-expansions named E_{p,q} and e_{p,q} of the exponential function x ↦ e^{x} are given as follows:

e_{p,q}(x) = ∑_{n=0}^{∞} \frac{p^{n(n−1)/2}}{[n]_{p,q}!} x^{n} =_{1} Φ_{0}((1, 0); −; (p, q), x),

(2.1)  E_{p,q}(x) = ∑_{n=0}^{∞} \frac{q^{n(n−1)/2}}{[n]_{p,q}!} x^{n} =_{1} Φ_{0}((0, 1); −; (p, q), −x).

We know that _{1}Φ_{0}((1, 0); −; (p, q), x)_{1}Φ_{0}((0, 1); −; (p, q), x) = 1, that is the following relation between (p, q)-exponential functions

(2.2)  e_{p,q}(x)E_{p,q}(−x) = 1

holds. We mention that these (p, q)-analogues of the classical exponential functions are valid for 0 < q < p ⩽ 1. Moreover E_{p,q}(x) and e_{p,q}(x) tend to e^{x} as p → 1^{−} and q → 1^{−} .

It is obvious by the (p, q)-derivative formula D_{p,q}f(x) = \frac{f(px)−f(qx)}{(p−q)x} , x ≠ 0 that

D_{p,q}E_{p,q}(x) = E_{p,q}(qx),
(2.3)  D_{p,q}E_{p,q}(ax) = aE_{p,q}(aqx).

Proposition 2.1. [13] The formula of (p, q)-integration by part is given by

∫_{a}^{b}f(px)D_{p,q}g(x) d_{p,q}x = f(b)g(b) − f(a)q(a) − ∫_{0}^{a} g(qx)D_{p,q}f(x) d_{p,q}x

Definition 2.1. For any n ∈ ℕ, we define a (p, q)-Gamma function by

Γ_{p,q}(n) = ∫_{0}^{∞} p^{(n−1)(n−2)/2} x^{n−1} E_{p,q}(−qx) d_{p,q}x.

Lemma 2.1. For any n ∈ ℕ, we have Γ_{p,q}(n + 1) = [n]_{p,q}!.

Proof. From (2.1) we have E_{p,q}(0) = 1 and from (2.2) we have

E_{p,q}(∞) = lim_{x→∞} E_{p,q}(x) = lim_{x→∞} e_{p,q}(−x) = lim_{x→∞} Φ_{0}((1, 0); −; (p, q), −x)

= lim_{x→∞} \frac{((p, 0); −; (p, q))_{∞}}{((p, −x); −; (p, q))_{∞}} = 0.

Also from (2.3) we can write

Γ_{p,q}(n + 1) = ∫_{0}^{∞} p^{n(n−1)/2} x^{n}E_{p,q}(−qx) d_{p,q}x
= − ∫_{0}^{∞} p^{n(n−1)/2} x^{n}D_{p,q}E_{p,q}(−x) d_{p,q}x.

By Proposition 2.1 using (p, q)-integration by parts for f(x) = x^{n} and g(x) = E_{p,q}(−x), we have

Γ_{p,q}(n + 1) = \frac{[n]_{p,q}}{p^{n−1}} ∫_{0}^{∞} p^{n(n−1)/2} x^{n−1} E_{p,q}(−qx) d_{p,q}x
= [n]_{p,q} ∫_{0}^{∞} p^{(n−1)(n−2)/2} x^{n−1} E_{p,q}(−qx) d_{p,q}x = [n]_{p,q}Γ_{p,q}(n).

Thus, we have

Γ_{p,q}(n + 1) = [n]_{p,q}Γ_{p,q}(n) = [n]_{p,q}[n − 1]_{p,q}Γ_{p,q}(n − 1) = [n]_{p,q}!.  □

An alternate form of (p, q)-Gamma function without integral expression for n nonnegative integer, is given in [12] by

Γ_{p,q}(n + 1) = \frac{(p ⊖ q)_{p,q}^{n}}{(p − q)^{n}} = [n]_{p,q}!, 0 < q < p.

3. (p, q) -Szász–Durrmeyer Operators and Moments

In order to introduce a (p, q) Durrmeyer variant for Szasz–Mirakyan operators, we present a construction due to Acar [1]. The (p, q)-analogue of Szász operators for x ∈ [0, ∞) and 0 < q < p ⩽ 1 defined by in the following way

(3.1)  S_{n,p,q} (f; x) = ∑_{k=0}^{n} s_{n,k}^{p,q} (x)f( \frac{[k]_{p,q}}{q^{k−2} [n]_{p,q}}) ,

where

s_{n,k}^{p,q} (x) = \frac{1}{E_{p,q}([n]_{p,q}x)} \frac{q^{k(k−1)/2}}{[k]_{p,q}!} ([n]_{p,q}x)^{k}.

In case p = 1, we get the q-Szász operators [2]. If p = q = 1, we get at once the well known Szász operators.

Lemma 3.1. [1] For x ∈ x ∈ [0, ∞), 0 < q < p ⩽ 1, we have

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

The Szász operators defined by (3.1) are discrete operators. The integral modification of these operators was proposed in [7]. Different variants and q-analogues in [3] and [6]. As an application of the (p, q)-Gamma function, we introduce below the Durrmeyer type (p, q) variant of the Szász operators as

Definition 3.1. The (p, q)-analogue of Szász–Durrmeyer operator for x ∈ [0, ∞) and 0 < q < p ⩽ 1 is defined by

S̃_{n,p,q} (f; x) = [n]_{p,q} ∑_{k=0}^{∞} s_{n,k}^{p,q} (x)∫_{0}^{∞}p^{k(k−1)/2} \frac{([n]_{p,q}t)^{k}}{[k]_{p,q}!}E_{p,q}(−q[n]_{p,q}t)f(q^{1−k} p^{k}t) d_{p,q}t

where s_{n,k}^{p,q} (x) is defined in (3.1).

It may be remarked here that for p = q = 1 these operators reduce to the Szász–Durrmeyer operators.

Lemma 3.2. For x ∈ x ∈ [0, ∞), 0 < q < p ⩽ 1, we have

(1) S̃_{n,p,q} (1; x) = 1

(2) S̃_{n,p,q} (t; x) = \frac{q}{[n]_{p,q}} + px

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

Proof. Using Definition 2.1, Lemmas 2.1 and 3.1, we have

S̃_{n,p,q} (1; x) = [n]_{p,q} ∑_{k=0}^{∞} s_{n,k}^{p,q} (x) ∫_{0}^{∞} p^{k(k−1)/2} \frac{([n]_{p,q}t)^{k}}{[k]_{p,q}!}E_{p,q}(−q[n]_{p,q}t) d_{p,q}t

= ∑_{k=0}^{∞} s_{n,k}^{p,q} (x)\frac{Γ_{p,q}(k + 1)}{[k]_{p,q}!} = 1

and next using [k + 1]_{p,q} = q^{k} + p[k]_{p,q}, we have

S̃_{n,p,q} (t; x) = ∑_{k=0}^{∞} s_{n,k}^{p,q} (x) ∫_{0}^{∞} p^{k(k−1)/2} q^{1−k} p^{k} \frac{([n]_{p,q}t)^{k+1}}{[k]_{p,q}!} E_{p,q}(−q[n]_{p,q}t) d_{p,q}t

= ∑_{k=0}^{∞} s_{n,k}^{p,q} (x)q^{1−k} \frac{Γ_{p,q}(k + 2)}{[k]_{p,q}![n]_{p,q}}= ∑_{k=0}^{∞} s_{n,k}^{p,q} (x)q^{1−k} \frac{[k + 1]_{p,q}}{[n]_{p,q}}

= ∑_{k=0}^{∞} s_{n,k}^{p,q} (x)q^{1−k} \frac{(q^{k} + p[k]_{p,q})}{[n]_{p,q}}

SZÁSZ DURRMEYER OPERATORS 215

= \frac{q}{[n]_{p,q}}S_{n,p,q} (1; x) + \frac{p}{q}S_{n,p,q} (t; x) = \frac{q}{[n]_{p,q}} + px

and

S̃_{n,p,q} (t^{2}; x) = \frac{1}{[n]_{p,q}} ∑_{k=0}^{∞} s_{n,k}^{p,q} (x)

∫_{0}^{∞} p^{k(k−1)/2} q^{2−2k} p^{2k} \frac{([n]_{p,q}t)^{k+2}}{[k]_{p,q}!} E_{p,q}(−q[n]_{p,q}t) d_{p,q}t

= \frac{1}{[n]_{p,q}} ∑_{k=0}^{∞} s_{n,k}^{p,q} (x)q^{2−2k}

∫_{0}^{∞} p^{(k+1)(k+2)/2} p^{−1} \frac{([n]_{p,q}t)^{k+2}}{[k]_{p,q}!} E_{p,q}(−q[n]_{p,q}t) d_{p,q}t

= \frac{1}{p[n]_{p,q}^{2}} ∑_{k=0}^{∞} s_{n,k}^{p,q} (x)q^{2−2k} \frac{Γ_{p,q}(k + 3)}{[k]_{p,q}!}

= \frac{1}{p[n]_{p,q}^{2}} ∑_{k=0}^{∞} s_{n,k}^{p,q} (x)q^{2−2k} [k + 2]_{p,q}[k + 1]_{p,q}

= \frac{1}{p[n]_{p,q}^{2}} ∑_{k=0}^{∞} s_{n,k}^{p,q} (x)q^{2−2k} (p^{3}[k]_{p,q}^{2} + q^{k}(p[2]_{p,q} + p^{2})[k]_{p,q} + q^{2k}[2]_{p,q})

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

= \frac{p^{2}}{q^{2}}(pqx^{2} + \frac{q^{2}}{[n]_{p,q}}x) + \frac{([2]_{p,q} + p)qx}{[n]_{p,q}} + \frac{[2]_{p,q}q^{2}}{p[n]_{p,q}^{2}}

= \frac{p^{3}}{q}x^{2} + \frac{[2]_{p,q}^{2}x}{[n]_{p,q}}+ \frac{[2]_{p,q}q^{2}}{p[n]_{p,q}^{2}}.  □

Remark 3.1. For 0 < q < p ⩽ 1 we may write

S̃_{n,p,q} ((t − x), x) = \frac{q}{[n]_{p,q}} + (p − 1)x,
(3.2)  S̃_{n,p,q} ((t − x)^{2}, x) = \frac{(p^{3} − 2pq + q)x^{2}}{q} + \frac{([2]_{p,q}^{2} − 2q)x}{[n]_{p,q}} + \frac{[2]_{p,q}q^{2}}{p[n]_{p,q}^{2}}.

4. Quantitative Estimate

By C_{B}[0, ∞) we denote the class of all real valued continuous and bounded functions on [0, ∞). The norm ∥.∥_{B} is defined by

∥f∥_{C_{B}} = sup_{x∈[0,∞)} |f(x)|.

For f ∈ C_{B} the Steklov mean is defined as

(4.1)  f_{h}(x) = \frac{4}{h^{2}} ∫0^{\frac{h}{2}} ∫0^{\frac{h}{2}} [2f(x + u + v) − f(x + 2(u + v))] du dv

By simple computation, it is observed that

(i) ∥f_{h} − f∥_{C_{B}} ⩽ ω_{2}(f, h). (ii) If f is continuous, then f′_{h}, f′′∈ C_{B} and

∥f′_{h}∥_{C_{B}} ⩽ \frac{5}{h} ω(f, h), ∥f′′_{h}∥_{C_{B}} ⩽ \frac{9}{h^{2}} Lω_{2}(f, h),

where the first and second order moduli of continuity are respectively defined by

ω(f, δ) = sup_{\begin{matrix}x,u,v⩾0 \\ |u−v|⩽δ \end{matrix}} |f(x + u) − f(x + v)|,

ω_{2}(f, δ) = sup_{\begin{matrix}x,u,v⩾0 \\ |u−v|⩽δ \end{matrix}} |f(x + 2u) − 2f(x + u + v) + f(x + 2v)|, δ ⩾ 0.

Theorem 4.1. Let q ∈ (0, 1) and p ∈ (q, 1] The operator S̃_{n,p,q} maps space C_{B} into C_{B} and ∥S̃_{n,p,q} (f)∥_{C_{B}} ⩽ ∥f∥_{C_{B}} .

Proof. Let q ∈ (0, 1) and p ∈ (q, 1]. From Lemma 3.2 we have

|S̃_{n,p,q} (f, x)|

⩽ [n]_{p,q} ∑_{k=0}^{∞} s_{n,k}^{p,q} (x) ∫_{0}^{∞} p^{k(k−1)/2} \frac{([n]_{p,q}t)^{k}}{[k]_{p,q}!}E_{p,q}(−q[n]_{p,q}t)|f(q^{1−k} p^{k}t)|d_{p,q}t

⩽ sup_{x∈[0,∞)} |f(x)[n]_{p,q} ∑_{k=0}^{∞} s_{n,k}^{p,q} (x) ∫_{0}^{∞} p^{k(k−1)/2} \frac{([n]_{p,q}t)^{k}}{[k]_{p,q}!}E_{p,q}(−q[n]_{p,q}t) d_{p,q}t

= sup_{x∈[0,∞)} |f(x)|S̃_{n,p,q} (1, x) = ∥f∥_{C_{B}} .  □

We are going to study the degree of approximation in terms of the first and second order moduli of continuity.

Theorem 4.2. Let q ∈ (0, 1) and p ∈ (q, 1]. If f ∈ C_{B}[0, ∞), then

|S̃_{n,p,q} (f, x) − f(x)|

⩽ 5ω( f, \frac{1}{\sqrt{[n]_{p,q}}}) ( \frac{q}{\sqrt{[n]_{p,q}}} + \sqrt{[n]_{p,q}}(p − 1)x)

+ \frac{9}{2}ω_{2}( f, \frac{1}{\sqrt{[n]_{p,q}}}) [2 + \frac{(p^{3} − 2pq + q)[n]_{p,q}x^{2}}{q} + ([2]_{p,q}^{2} − 2q)x + \frac{[2]_{p,q}q^{2}}{p[n]_{p,q}}].

Proof. For x ⩾ 0 and n ∈ ℕ and using the Steklov mean f_{h} defined by (4.1), we can write

|S̃_{n,p,q} (f, x) − f(x)| ⩽ S̃_{n,p,q} (|f − f_{h}|, x) + |S̃_{n,p,q} (f_{h} − f_{h}(x), x)| + |f_{h}(x) − f(x)|.

First by Theorem 4.1 and property (i) of the Steklov mean we have

S̃_{n,p,q} (|f − f_{h}|, x) ⩽ ∥S̃_{n,p,q} (f − f_{h})∥_{C_{B}} ⩽ ∥f − f_{h}∥_{C_{B}} ⩽ ω_{2}(f, h).

Since S̃_{n,p,q} is a linear positive operator we get

|S̃_{n,p,q} (f_{h} − f_{h}(x), x)| ⩽ |f′_{h}(x)|S̃_{n,p,q} (t − x, x) + \frac{1}{2}∥f′′∥_{C_{B}} S̃_{n,p,q} ((t − x)^{2}, x).

By Lemma 3.2, we have

|S̃_{n,p,q} (f_{h} − f_{h}(x), x)| ⩽ \frac{5}{h}ω(f, h)( \frac{q}{[n]_{p,q}} + (p − 1)x)

+ \frac{9}{2h^{2}} ω_{2}(f, h)S̃_{n,p,q} ((t − x)^{2}, x),

where S̃_{n,p,q} ((t − x)^{2}, x) is given by (3.2). For x ⩾ 0, h > 0 and choosing h = \sqrt{1/[n]_{p,q}}, we get the desired result. □

Remark 4.1. For q ∈ (0, 1) and p ∈ (q, 1] it is seen that lim_{n→∞} [n]_{p,q} = 1/(q−p). In order to consider the convergence of (p, q)-Szász–Durrmeyer operators, we assume p = (p_{n}) and q = (q_{n}) such that 0 < q_{n} < p_{n} ⩽ 1 and for n sufficiently large p_{n} → 1, q_{n} → 1, p_{n}^{n} → a, q_{n}^{n} → b, so that [n]_{p_{n} ,q_{n}} → ∞. Such a sequence can always be constructed for example, we can take p_{n} = 1 − 1/2n and q_{n} = 1 − 1/n. Clearly lim_{n→∞} p_{n}^{n} = e^{−1/2} , lim_{n→∞} q_{n}^{n} = e^{−1} and lim_{n→∞} [n]_{p_{n} ,q_{n}} = ∞.

5. Direct Estimates

Let us denote by H_{x^{2}} [0, ∞) the set of all functions f defined on the positive real axis satisfying the condition |f(x)| ⩽ M_{f} (1 + x^{2}), where M_{f} is an absolute constant depending on f. By C_{x^{2}} [0, ∞), we mean the subspace of all continuous functions belonging to H_{x^{2}} [0, ∞). Also, let C_{x^{2}}^{∗} [0, ∞) denote the subspace of all functions f ∈ C_{x^{2}} [0, ∞), for which lim_{|x|→∞} \frac{f(x)}{1+x^{2}} is finite. The class C_{x^{2}}^{∗} [0, ∞) is endowed with the norm

∥f∥_{x^{2}} = sup_{x∈[0,∞)} \frac{|f(x)|}{1 + x^{2}} .

We discuss below the weighted approximation theorem, where the approximation formula is valid for the positive real axis (see [4]).

Theorem 5.1. Let p = p_{n} and q = q_{n} satisfies 0 < q_{n} < p_{n} ⩽ 1 and for n sufficiently large p_{n} → 1, q_{n} → 1 and q_{n}^{n} → a and p_{n}^{n} → b. For each f ∈ C_{x^{2}}^{∗} [0, ∞), we have lim_{n→∞} ∥S̃_{n,p_{n} ,q_{n}} (f) − f∥_{x^{2}} = 0.

Proof. Using Korovkin’s theorem, it is sufficient to verify the following three conditions

(5.1)  lim_{n→∞} ∥S̃_{n,p_{n} ,q_{n}} (t^{ν}, x) − x^{ν}∥_{x^{2}} = 0, ν = 0, 1, 2.

Since S̃_{n,p_{n} ,q_{n}} (1, x) = 1 the first condition of (5.1) is fulfilled for ν = 0.

For n ∈ ℕ, we can write,

∥S̃_{n,p_{n} ,q_{n}} (t, x) − x∥_{x^{2}} ⩽ \frac{q_{n}}{[n]_{p_{n} ,q_{n}}} + (p_{n} − 1) sup_{x∈[0,∞)} \frac{x}{1 + x^{2}}

∥S̃_{n,p_{n} ,q_{n}} (t^{2}, x) − x^{2}∥_{x^{2}} ⩽ ( \frac{p_{n}^{3}}{q_{n}} − 1) sup_{x∈[0,∞)} \frac{x^{2}}{1 + x^{2}} + \frac{[2]_{p_{n} ,q_{n}}^{2}}{[n]_{p_{n} ,q_{n}}} sup_{x∈[0,∞)} \frac{x}{1 + x^{2}}

+ \frac{[2]_{p_{n} ,q_{n}} q_{n}^{2}}{p_{n}[n]_{p_{n} ,q_{n}}^{2}} sup_{x∈[0,∞)} \frac{1}{1 + x^{2}} ,

which implies that for v = 1, 2 we have lim_{n→∞} ∥S̃_{n,p_{n} ,q_{n}} (t^{v}, x) − x^{v}∥_{x^{2}} = 0. □

We give the following theorem to approximate all functions in C_{x^{2}} [0, ∞).

Theorem 5.2. Let p = p_{n} and q = q_{n} satisfies 0 < q_{n} < p_{n} ⩽ 1 and for n sufficiently large p_{n} → 1, q_{n} → 1 and q_{n}^{n} → a and p_{n}^{n} → b. For each f ∈ C_{x^{2}} [0, ∞) and α > 0, we have

lim_{n→∞} sup_{x∈[0,∞)} \frac{|S̃_{n,p_{n} ,q_{n}} (f, x) − f(x)|}{(1 + x^{2})^{1+α}} = 0.

Proof. For any fixed x_{0} > 0,

sup_{x∈[0,∞)} \frac{|S̃_{n,p_{n} ,q_{n}} (f, x) − f(x)|}{(1 + x^{2})^{1+α}}

= sup_{x⩽x_{0}} \frac{|S̃_{n,p_{n} ,q_{n}} (f, x) − f(x)|}{(1 + x^{2})^{1+α}} + sup_{x⩾x_{0}} \frac{|S̃_{n,p_{n} ,q_{n}} (f, x) − f(x)|}{(1 + x^{2})^{1+α}}

⩽ ∥S̃_{n,p_{n} ,q_{n}} (f) − f∥_{C[0,a]}

+ ∥f∥_{x^{2}} sup_{x⩾x_{0}} \frac{|S̃_{n,p_{n} ,q_{n}} (1 + t^{2}, x)∥}{(1 + x^{2})^{1+α}} + sup_{x⩾x_{0}} \frac{|f(x)|}{(1 + x^{2})^{1+α}} .

By Lemma 3.2 and the well known Korovkin theorem, the first term of the above inequality tends to zero for a sufficiently large n. By Lemma 3.2, for any fixed x_{0} > 0, it is easily seen that sup_{x⩾x_{0}} \frac{|S̃_{n,pn ,qn} (1+t^{2} ,x)|}{(1+x^{2} )^{1+α}} tends to zero as n → ∞. We can choose x_{0} > 0 so large that the last part of above inequality can be made small enough. This completes the proof of the theorem. □

Now we establish some point-wise estimates of the rate of convergence of (p, q)- Szász–Durrmeyer operators. First, we give the relationship between the local smoothness of f and local approximation. A function f ∈ C[0, ∞) is is said to satisfy the Lipschitz condition Lip_{α} on D, α ∈ (0, 1], D ⊂ [0, ∞) if

(5.2)  |f(t) − f(x)| ⩽ M_{f} |t − x|^{α} , t ∈ [0, ∞) and x ∈ D,

where M_{f} is a constant depending only α and f.

Theorem 5.3. Let f ∈ Lip_{α} on D, D ⊂ [0, ∞) and α ∈ (0, 1]. We have

|S̃_{n,p,q} (f, x) − f(x)| ⩽ ( \frac{(p^{3} − 2pq + q)x^{2}}{q} + \frac{([2]_{p,q}^{2} − 2q)x}{[n]_{p,q}} + \frac{[2]_{p,q}q^{2}}{p[n]_{p,q}^{2}}) ^{α/2}

+2d^{α} (x; D)

where d(x; D) represents the distance between x and D.

Proof. For x_{0} ∈ D̄, the closure of the set D in [0, ∞), we have

|f(t) − f(x)| ⩽ |f(t) − f(x_{0})| + |f(x_{0}) − f(x)|, x ∈ [0, ∞).

Using (5.2) we get

(5.3)  |S̃_{n,p,q} (f, x) − f(x)| ⩽ S̃_{n,p,q} (|f(t) − f(x_{0})|, x) + |f(x_{0}) − f(x)|
⩽ M_{f} S̃_{n,p,q} (|t − x_{0}|^{α} , x) + M_{f} |x_{0} − x|^{α} .

Then, with Hölder’s inequality with p := \frac{2}{α} and \frac{1}{r} := 1 − \frac{1}{p}, we have

(5.4)  S̃_{n,p,q} (|t − x|^{α} , x) ⩽ (S̃_{n,p,q} (|t − x|^{2}, x))^{\frac{α}{2}} (S̃_{n,p,q} (1, x))^{1− \frac{α}{2}} .

Also S̃_{n,p,q} is monotone

S̃_{n,p,q} (|t − x_{0}|^{α} , x) ⩽ (S̃_{n,p,q} (|t − x|^{α} , x))^{\frac{α}{2}} + |x_{0} − x|^{α} .

Using (5.3), (5.4) and (3.2), we get the desired result. □

Now, we give a local direct estimate for (p, q)-Szász–Durrmeyer operators using the Lipschitz-type maximal function of order α introduced by Lenze [8] as

(5.5)  ω̃_{α} (f, x) = sup_{t≠x, t∈[0,∞)}  \frac{|f(t) − f(x)|}{|t − x|^{α}} , x ∈ [0, ∞) and α ∈ (0, 1].

Theorem 5.4. Let f ∈ Lip_{α} on D and f ∈ C_{B}[0, ∞). Then for all x ∈ [0, ∞),

we have

|S̃_{n,p,q} (f, x) − f(x)| ⩽ ω̃_{α} (f, x)( \frac{(p^{3} − 2pq + q)x^{2}}{q} + \frac{([2]_{p,q}^{2} − 2q)x}{[n]_{p,q}} + \frac{[2]_{p,q}q^{2}}{p[n]_{p,q}^{2}}) ^{\frac{α}{2}}

Proof. From (5.5) we have

|f(t) − f(x)| ⩽ ω̃_{α} (f, x)|t − x|^{α} ,
|S̃_{n,p,q} (f, x) − f(x)| ⩽ S̃_{n,p,q} (|f(t) − f(x)|, x)
⩽ ω̃_{α} (f, x)S̃_{n,p,q} (|t − x|^{α} , x).

Applying Hölder’s inequality with p := \frac{2}{α} and \frac{1}{r} := 1 − \frac{1}{p}, we have

|S̃_{n,p,q} (f, x) − f(x)| ⩽ ω̃_{α} (f, x)S̃_{n,p,q} ((t − x)^{2}, x)^{\frac{α}{2}} .

Using (3.2), we have our assertion. □

References

1. T. Acar, (p, q)-Generalization of Szász–Mirakyan operators, Math. Methods Appl. Sci. 39(10) (2016), 2685—2695.

2. A. Aral, V. Gupta, The q-derivative and applications to q-Szász Mirakyan operators, Calcolo 43(3) (2006), 151–170.

3. A. Aral, V. Gupta, R. P. Agarwal, Applications of q Calculus in Operator Theory, Springer, 2013.

4. A. D. Gadzhiev, Theorems of the type of P. P. Korovkin type theorems, Math. Zametki 20(5) (1976), 781–786.

5. V. Gupta, A. Aral, Bernstein Durrmeyer operators based on two paremeters, Facta Univ., Ser. Math. Inf. 31(1) (2016), 79–95.

6. V. Gupta, R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, 2014.

7. H. S. Kasana, G. Prasad, P. N. Agrawal, Modified Szász operators, Pergamon Press Oxford, 1988, 29–41.

8. B. Lenze, On Lipschitz-type maximal functions and their smoothness spaces, Indag. Math. 50(1) (1988), 53–63.

9. M. Mursaleen, K. J. Ansari, A. Khan, Some approximation results by (p, q)-analogue of Bernstein–Stancu operators, Appl.Math. Comput. 264 (2015), 392–402.

10. , On (p, q)-analogue of Bernstein operators, Appl. Math. Comput. 266 (2016), 874– 882.

11. M. Mursaleen, Md. Nasiruzzaman, A. Khan, K. J. Ansari, Some approximation results on Bleimann–Butzer–Hahn operators defined by (p, q)-integers, Filomat 30(3) (2016), 639–648.

12. P. N. Sadjang, On the (p, q)-Gamma and the (p, q)-Beta functions, arXiv 1506.07394v1.

13. , On the fundamental theorem of (p, q)-Calculus and some (p, q)-Taylor formulas, arXiv:1309.3934.

14. V. Sahai, S. Yadav, Representations of two parameter quantum algebras and (p, q)-special functions, J. Math. Anal. Appl. 335 (2007), 268–279.

Department of Mathematics (Received 25 02 2016) Kirikkale University (Revised 27 11 2016) Yahsihan Kirikkale Turkey

aliaral73@yahoo.com

Department of Mathematics Netaji Subhas Institute of Technology Sector 3 Dwarka New Delhi India

vijaygupta2001@hotmail.com