International Journal of Analysis and Applications ISSN 2291-8639 Volume 11, Number 1 (2016), 28-39 http://www.etamaths.com ON CHLODOWSKY VARIANT OF (p,q) KANTOROVICH-STANCU-SCHURER OPERATORS VISHNU NARAYAN MISHRA1,2,∗ AND SHIKHA PANDEY1 Abstract. In the present paper, we introduce the Chlodowsky variant of (p,q) Kantorovich-Stancu- Schurer operators on the unbounded domain which is a generalization of (p,q) Bernstein-Stancu- Kantorovich operators. We have also derived its Korovkin type approximation properties and rate of convergence. 1. Introduction and preliminaries Approximation theory has an important role in mathematical research because of its great potential for applications. Korovkin gave his famous approximation theorem in 1950, since then the study of the linear methods of approximation given by sequences of positive and linear operators became a deep- rooted part of approximation theory. Considering it, various operators as Bernstein, Szász, Baskakov etc. and their generalizations are being studied. In recent years, many results about the generalization of positive linear operators have been obtained by several mathematicians ([6], [8]-[12], [21]). In last two decades, the applications of q-calculus has played an important role in the area of approximation theory, number theory and theoretical physics. In 1987, Lupas, and in 1997, Phillips introduced a sequence of Bernstein polynomials based on q-integers and investigated its approximation properties. Several researchers obtained various other generalizations of operators based on q-calculus(See [3], [17]-[19]). Recently, Mursaleen et al. applied (p,q)-calculus in approximation theory and introduced first (p,q)- analogue of Bernstein operators. They investigated uniform convergence of the operators and order of convergence, obtained Voronovskaja theorem as well. Also, (p,q)-analogue of Bernstein opera- tors, Bernstein-Stancu operators and Bernstein-Schurer operators, Kontorovich Bernstein Schurer, and Bleimann-Butzer-Hahn operators were introduced in ([13]-[16]), respectively. Further, Acar [1] have studied recently, (p,q)-Generalization of SzászMirakyan operators. In the present paper, we introduce the Chlodowsky variant of (p,q) Kantorovich-Stancu-Schurer op- erators on the unbounded domain. We begin by recalling certain notations of (p,q)-calculus. For 0 < q < p ≤ 1, the (p,q) integer [n]p,q is defined by [n]p,q := pn −qn p−q . (p,q) factorial is expressed as [n]p,q! = [n]p,q[n− 1]p,q[n− 2]p,q . . . 1. (p,q) binomial coefficient is expressed as[ n k ] p,q := [n]p,q! [k]p,q![n−k]p,q! . 2010 Mathematics Subject Classification. 41A25, 41A36, 41A10, 41A30. Key words and phrases. (p,q)-integers; (p,q) Bernstein operators; Chlodowsky polynomials; (p,q) Bernstein-Kantorovich operators; modulus of continuity; linear positive operator; Korovkin type approximation theorem; rate of convergence. c©2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 28 ON CHLODOWSKY VARIANT OF (p,q) KANTOROVICH-STANCU-SCHURER OPERATORS 29 (p,q) binomial expansion as (ax + by)np,q := n∑ k=0 [ n k ] p,q an−kbkxn−kyk. (x + y)np,q := (x + y)(px + qy)(p 2x + q2y) . . . (pn−1x + qn−1y). The definite integrals of the function f are given by∫ a 0 f(x)dp,qx = (q −p)a ∞∑ k=0 pk qk+1 f ( pk qk+1 a ) , ∣∣∣∣pq ∣∣∣∣ < 1, and ∫ a 0 f(x)dp,qx = (p−q)a ∞∑ k=0 qk pk+1 f ( qk pk+1 a ) , ∣∣∣∣pq ∣∣∣∣ > 1. Further (p,q) analysis can be found in [2]. In 1932, Chlodowsky [7] presented a generalization of Bernstein polynomials on an unbounded set, known as Bernstein-Chlodowsky polynomials given by, (1.1) Bn(f,x) = n∑ k=0 f( k n bn) ( n k )( x bn )k ( 1 − x bn )n−k , 0 ≤ x ≤ bn, where bn is an increasing sequence of positive terms with the properties bn → ∞ and bnn → 0 as n →∞. In 2015, Vedi and Özarslan [20] investigated Chlodowsky-type q-Bernstein-Stancu-Kantorovich opera- tors, and Wafi and Rao investigated (p,q) form of Kantorovich type Bernstein-Stancu-Schurer operator. Mursaleen and Khan [15] defined the Kantorovich type (p,q)-Bernstein-Schurer Operators, given by Tn,m(f; x,p,q) = n+m∑ k=0 [ n + m k ] p,q xk n+m−k−1∏ s=0 (ps −qsx) ∫ 1 0 f ( (1 − t)[k]p,q + [k + 1]p,qt [n + 1]p,q ) dp,qt k = 0, 1, 2, . . . ,n = 1, 2, 3, . . . where Lemma 1. (See [15]) For the Operators T (α,β) n,m , we have Tn,m(1; x,p,q) = 1, Tn,m(t; x,p,q) = (px + 1 −x)n+mp,q [2]p,q[n + 1]p,q + (p + 2q − 1)[n + m]p,q [2]p,q[n + 1]p,q x, Tn,m(t 2; x,p,q) = (p2x + 1 −x)n+mp,q [3]p,q[n + 1]2p,q + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q [n + 1]2p,q (px + 1 −x)n+m−1p,q x + { 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q [n + 1]2p,q x2. 2. Construction of the operators We construct the Chlodowsky variant of (p,q) Kontorovich-Stancu-Schurer operators as (2.1) K (α,β) n,m (f; x,p,q) = n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k ∫ 1 0 f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) dp,qt, where n ∈ N, m,α,β ∈ N0 with 0 ≤ α ≤ β, 0 ≤ x ≤ bn, 0 < q < p ≤ 1. Obviously, K (α,β) n,m is a linear and positive operator. To begin with, we obtain the following important lemma. 30 MISHRA AND PANDEY Lemma 2. Let K (α,β) n,m (f; x,p,q) be given by (2.1). The first few moments of the operators are (i) K(α,β)n,m (1; x,p,q) = 1, (ii) K(α,β)n,m (t; x,p,q) = ( 1 [n + 1]p,q + β )( αbn + (p x bn + 1 − x bn )n+mp,q [2]p,q bn + (p + 2q − 1)[n + m]p,q [2]p,q x ) , (iii) K(α,β)n,m (t 2; x,p,q) = ( 1 [n + 1]p,q + β )2[( α2 + 2α [2]p,q ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q ) b2n + ( 2α [2]p,q (p + 2q − 1) + { 1 + 2q [2]p,q + q2 − 1 [3]p,q }( p x bn + 1 − x bn )n+m−1 p,q ) [n + m]p,qbnx + { 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,qx2 ] , (iv) K(α,β)n,m ((t−x); x,p,q) = [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x (v) K(α,β)n,m ((t−x) 2; x,p,q) = [ α2 ([n + 1]p,q + β)2 + 2α [2]p,q([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q([n + 1]p,q + β)2 ] b2n + [ 2α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m−1 p,q − 2α ([n + 1]p,q + β) − 2(p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) ] bnx + [{ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q ([n + 1]p,q + β)2 − 2 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 1 ] x2. Proof. From operator 2.1, K (α,β) n,m (t u; x,p,q) = n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn )u dp,qt = n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k [n + 1]up,qbun ([n + 1]p,q + β)u × ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q )u dp,qt = [n + 1]up,qb u n ([n + 1]p,q + β)u n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) u∑ i=0 ( u i )( α [n + 1]p,q )u−i × ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt [n + 1]p,q )u dp,qt = [n + 1]up,qb u n ([n + 1]p,q + β)u u∑ i=0 ( u i )( α [n + 1]p,q )u−i n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) × ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt [n + 1]p,q )i dp,qt K (α,β) n,m (t u; x,p,q) = [n + 1]up,qb u n ([n + 1]p,q + β)u u∑ i=0 ( u i )( α [n + 1]p,q )u−i Tn,m(t i; x bn ,p,q).(2.2) Thus for u=0,1,2 we get K(α,β)n,m (1; x,p,q) = Tn,m(1; x bn ,p,q), ON CHLODOWSKY VARIANT OF (p,q) KANTOROVICH-STANCU-SCHURER OPERATORS 31 K(α,β)n,m (t; x,p,q) = [n + 1]p,qbn [n + 1]p,q + β 1∑ i=0 ( 1 i )( α [n + 1]p,q )1−i Tn,m(t i; x bn ,p,q) = ( [n + 1]p,q [n + 1]p,q + β ) bn { α [n + 1]p,q + Tn,m(t; x bn ,p,q) } , K(α,β)n,m (t 2; x,p,q) = [n + 1]2p,qb 2 n ([n + 1]p,q + β)2 2∑ i=0 ( 2 i )( α [n + 1]p,q )2−i Tn,m(t i; x bn ,p,q) = [n + 1]2p,qb 2 n ([n + 1]p,q + β)2 [( α [n + 1]p,q )2 +2 ( α [n + 1]p,q ) Tn,m(t; x bn ,p,q) + Tn,m(t 2; x bn ,p,q) ] . Using Lemma 1 and in view of the above relations we get the statements (i), (ii) and (iii). Using linear property of operators, we have K(α,β)n,m ((t−x); x,p,q) = K (α,β) n,m (t; x,p,q) −xK (α,β) n,m (1; x,p,q) = [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x. Hence, we get (iv). Similar calculations give K(α,β)n,m ((t−x) 2; x,p,q) = K(α,β)n,m (t 2; x,p,q) − 2xK(α,β)n,m (t; x,p,q) + x 2K(α,β)n,m (1; x,p,q). Then we obtain, K(α,β)n,m ((t−x) 2; x,p,q) = [ α2 ([n + 1]p,q + β)2 + 2α [2]p,q([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q([n + 1]p,q + β)2 ] b2n + [ 2α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m−1 p,q − 2α ([n + 1]p,q + β) − 2(p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) ] bnx + [{ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q ([n + 1]p,q + β)2 − 2 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 1 ] x2. This proves (v). � 3. Korovkin-type approximation theorem Assume Cρ is the space of all continuous functions f such that |f(x)| ≤ Mρ(x), −∞ < x < ∞. Then Cρ is a Banach space with the norm ‖f‖ρ = sup −∞ 0. Theorem 2. (See [5]) Conditions (1),(2),(3) of above theorem implies that lim n→∞ ‖Unf −f‖ρ = 0 for any function f belonging to the subset C0ρ := {f ∈ Cρ : lim|x|→∞ |f(x)| ρ(x) is finite} . Consider the weight function ρ(x) = 1 + x2 and operators: Uα,βn,m(f; x,p,q) = { Kα,βn,m(f; x,p,q) if x ∈ [0,bn], f(x) if x ∈ [0,∞)/[0,bn]. Thus for f ∈ C1+x2, we have ‖Uα,βn,m(f; ·,p,q)‖≤ sup x∈[0,bn] |Uα,βn,m(f; x,p,q)| 1 + x2 + sup bn 0). For any � > 0, we have∣∣∣∣f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) −f(x) ∣∣∣∣ < � + 2Mδ2 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 , where x ∈ [0,bn] and δ = δ(�) are independent of n. Now since we know, K (α,β) n,m ((t−x)2; x,pn,qn) = n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps−qs x bn ) ( x bn )k ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn )2 dt. we can conclude by Theorem 3, sup 0≤x≤bn ∣∣∣Kα,βn,m(f; x,pn,qn) −f(x)∣∣∣ ≤ � + 2M δ2 b2n ([ α2 ([n + 1]p,q + β)2 + 2α [2]p,q([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q([n + 1]p,q + β)2 ] + ∣∣∣∣∣2α(p + 2q − 1)[n + m]p,q[2]p,q([n + 1]p,q + β)2 + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m−1 p,q − 2α ([n + 1]p,q + β) − 2(p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) ∣∣∣∣∣ + ∣∣∣∣∣ { 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q ([n + 1]p,q + β)2 − 2 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 1 ∣∣∣∣∣ ) . Since b2n [n]p,q = 0 as n →∞, we have the desired result. � 4. Rate of Convergence We will find the rate of convergence in terms of the Lipschitz class LipM (γ), for 0 < γ ≤ 1. Assume that CB[0,∞) denote the space of bounded continuous functions on [0,∞). A function f ∈ CB[0,∞) belongs to LipM (γ) if |f(t) −f(x)| ≤ M|t−x|γ, t,x ∈ [0,∞) is satisfied. 34 MISHRA AND PANDEY Theorem 5. Let f ∈ LipM (γ), then Kα,βn,m(f; x,p,q) ≤ M(µn,p,q(x)) γ/2, where µn,p,q(x) = K α,β n,m((t−x)2; x,p,q). Proof. Since f ∈ LipM (γ), and the operator Kα,βn,m(f; x,p,q) is linear and monotone, |Kα,βn,m(f; x,p,q) −f(x)| = ∣∣∣∣∣ n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k × ∫ 1 0 ( f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) −f(x) ) dp,qt ∣∣∣∣∣ ≤ n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k × ∫ 1 0 ∣∣∣∣f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) −f(x) ∣∣∣∣dp,qt ≤ M n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k × ∫ 1 0 ∣∣∣∣(1 − t)[k]p,q + [k + 1]p,qt + α[n + 1]p,q + β bn −x ∣∣∣∣γ dp,qt . Applying Hölder’s inequality with the values p = 2 γ and q = 2 2−γ , we get following inequality,∫ 1 0 ∣∣∣∣(1 − t)[k]p,q + [k + 1]p,qt + α[n + 1]p,q + β bn −x ∣∣∣∣γ dp,qt ≤ {∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt }γ 2 {∫ 1 0 dp,qt }2−γ 2 = {∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt }γ 2 Using this, we get |Kα,βn,m(f; x,p,q) −f(x)| ≤ M n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k × {∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt }γ 2 = M n+m∑ k=0 {∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt }γ 2 wn,k(p,q; x), where wn,k(p,q; x) = [ n + m k ] p,q ∏n+m−k−1 s=0 (p s−qs x bn ) ( x bn )k . Again using Hölder’s inequality with p = 2 γ and q = 2 2−γ , we have |Kα,βn,m(f; x,p,q) −f(x)| ≤ M { n+m∑ k=0 ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt wn,k(p,q; x) }γ 2 { n+m∑ k=0 wn,k(p,q; x) }2−γ 2 ON CHLODOWSKY VARIANT OF (p,q) KANTOROVICH-STANCU-SCHURER OPERATORS 35 = M { n+m∑ k=0 wn,k(p,q; x) ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt }γ 2 = M(µn,p,q(x)) γ/2, where (µn,p,q(x)) γ/2 = Kα,βn,m((t−x)2; x,p,q). � In order to obtain rate of convergence in terms of modulus of continuity ω(f; δ), we assume that for any uniformly continuous f ∈ CB[0,∞) and x ≥ 0, modulus of continuity of f is given by (4.1) ω(f; δ) = max |t−x|≤δ t,x∈[0,∞) |f(t) −f(x)|. Thus it implies for any δ > 0 (4.2) |f(x) −f(y)| ≤ ω(f; δ) ( |x−y| δ + 1 ) , is satisfied. Theorem 6. If f ∈ CB[0,∞), we have |Kα,βn,m(f; x,p,q) −f(x)| ≤ 2ω(f; √ µn,p,q(x)), where ω(f; ·) is modulus of continuity of f and λn,p,q(x) be the same as in Theorem 5. Proof. Using triangular inequality, we get |Kα,βn,m(f; x,p,q) −f(x)| = ∣∣∣∣∣ n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) ( f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) −f(x) )∣∣∣∣∣ ≤ n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) ∣∣∣∣f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) −f(x) ∣∣∣∣ , Now using (4.2) and Hölder’s inequality, we get |Kα,βn,m(f; x,p,q) −f(x)| = n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn )  |(1−t)[k]p,q+[k+1]p,qt+α[n+1]p,q+β bn −x| δ + 1  ω(f; δ) ≤ ω(f; δ) n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) + ω(f; δ) δ n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) ∣∣∣∣(1 − t)[k]p,q + [k + 1]p,qt + α[n + 1]p,q + β bn −x ∣∣∣∣ = ω(f; δ) + ω(f; δ) δ { n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) × ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 }12 = ω(f; δ) + ω(f; δ) δ { Kα,βn,m((t−x) 2; x,p,q) }1/2 . Now choosing δ = µn,p,q(x) as in Theorem 5, we have |Kα,βn,m(f; x,p,q) −f(x)| ≤ 2ω(f; √ µn,p,q(x)). � 36 MISHRA AND PANDEY Now let us denote by C2B[0,∞) the space of all functions f ∈ CB[0,∞) such that f ′,f′′ ∈ CB[0,∞). Let ‖f‖ denote the usual supremum norm of f . The classical Peetre’s K-functional and the second modulus of smoothness of the function f ∈ CB[0,∞) are defined respectively as K(f,δ) := inf g∈C2 B [0,∞) [‖f −g‖ + δ‖g′′‖] and ω2(f,δ) = sup 0 0. It is known that [see [4], p. 177] there exists a constant A > 0 such that (4.3) K(f,δ) ≤ Aω2(f,δ). Theorem 7. Let x ∈ [0,bn] and f ∈ CB[0,∞). Then, for fixed p ∈ N0, we have |Kα,βn,m(f; x,p,q) −f(x)| ≤ Cω2(f, √ αn,p,q(x)) + ω(f,βn,p,q(x)) for some positive constant C, where αn,p,q(x) = [{ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q + (p + 2q − 1)2 [2]2p,q } [n + m]2p,q ([n + 1]p,q + β)2 − 4 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 2 ] x2 + [{ 1 + 2q [2]p,q + q2 − 1 [3]p,q + 2 (p + 2q − 1) [2]2p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q +4 α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 − 4 (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) − 4 α ([n + 1]p,q + β) ] bnx + [ (p2 x bn + 1 − x bn )n+mp,q [3]p,q + (p x bn + 1 − x bn )2n+2mp,q [2]2p,q + 4 α [2]p,q (p x bn + 1 − x bn )n+mp,q + 2α 2 ] b2n ([n + 1]p,q + β)2 ,(4.4) and (4.5) βn,p,q(x) = ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x ) . Proof. Consider an auxiliary operator K∗n,m(f; x,p,q) : CB[0,∞) → CB[0,∞) by (4.6) K∗n,m(f; x,p,q) := K α,β n,m(f; x,p,q) −f ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x ) + f(x). Then by Lemma 2 we get K∗n,m(1; x,p,q) = 1, K∗n,m((t−x); x,p,q) = 0. (4.7) For given g ∈ CB[0,∞), it follows by the Taylor formula that g(y) −g(x) = (y −x)g′(x) + ∫ y x (y −u)g′′(u) du. Taking into account 4.6 and using 4.7, we get |K∗n,m(g; x,p,q) −g(x)| = |K ∗ n,m(g(y) −g(x); x,p,q)| = ∣∣∣∣g′(x)K∗n,m((y −x); x,p,q) + K∗n,m (∫ y x (y −u)g′′(u) du; x,p,q )∣∣∣∣ = ∣∣∣∣K∗n,m (∫ y x (y −u)g′′(u) du; x,p,q )∣∣∣∣ Then by 4.6 |K∗n,m(g; x,p,q) −g(x)| = ∣∣∣∣∣K∗n,m (∫ y x (y −u)g′′(u) du; x,p,q ) − ∫ ( [2]p,qα+(p xbn +1− xbn )n+mp,q [2]p,q([n+1]p,q+β) bn+ (p+2q−1)[n+m]p,q [2]p,q([n+1]p,q+β) x ) x ON CHLODOWSKY VARIANT OF (p,q) KANTOROVICH-STANCU-SCHURER OPERATORS 37( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x−u ) g′′(u) du ∣∣∣∣∣ ≤ ∣∣∣∣∣K∗n,m (∫ y x (y −u)g′′(u) du; x,p,q )∣∣∣∣∣+ ∣∣∣∣∣ ∫ ( [2]p,qα+(p xbn +1− xbn )n+mp,q [2]p,q([n+1]p,q+β) bn+ (p+2q−1)[n+m]p,q [2]p,q([n+1]p,q+β) x ) x( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x−u ) g′′(u) du ∣∣∣∣∣. Since, ∣∣∣Kα,βn,m (∫ y x (y −u)g′′(u) du; x,p,q )∣∣∣≤‖g′′(x)‖ Kα,βn,m((y −x)2; x,p,q) and ∣∣∣∣∣ ∫ ( [2]p,qα+(p xbn +1− xbn )n+mp,q [2]p,q([n+1]p,q+β) bn+ (p+2q−1)[n+m]p,q [2]p,q([n+1]p,q+β) x ) x( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x−u ) g′′(u) du ∣∣∣∣∣ ≤ ‖g′′‖ [ [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x ]2 , we get |K∗n,m(g; x,p,q) −g(x)| ≤ ‖g ′′‖Kα,βn,m((y −x) 2; x,p,q) + ‖g′′‖ [ [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x ]2 . Hence Lemma 2 implies that |K∗n,m(g; x,p,q) −g(x)|(4.8) ≤‖g′′‖ [( α2 ([n + 1]p,q + β)2 + 2α [2]p,q([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q([n + 1]p,q + β)2 ) b2n + ( 2α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m−1 p,q − 2α ([n + 1]p,q + β) − 2(p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) ) bnx + ({ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q ([n + 1]p,q + β)2 − 2 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 1 ) x2 + ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x )2] .(4.9) Since K∗n,m(f; x,p,q) ≤ 3‖f‖, considering 4.4 and 4.5,for all f ∈ CB[0,∞) and g ∈ C2B[0,∞), we may write from 4.8 that |Kα,βn,m(f; x,p,q) −f(x)| ≤ |K ∗ n,m(f −g; x,p,q) − (f −g)(x)| + |K ∗ n,m(g; x,p,q) −g(x)| + ∣∣∣∣∣f ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x ) −f(x) ∣∣∣∣∣ ≤ 4‖f −g‖ + αn,p,q(x)‖g′‖ + ∣∣∣∣∣f ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x ) −f(x) ∣∣∣∣∣ ≤ 4‖f −g‖ + αn,p,q(x)‖g′‖ + ω(f,βn,p,q(x)), 38 MISHRA AND PANDEY which yields that |Kα,βn,m(f; x,p,q) −f(x)| ≤ 4K(f,αn,p,q(x)) + ω(f,betan,p,q(x)) ≤ Cω2(f, √ αn,p,q(x)) + ω(f,βn,p,q(x)), where αn,p,q(x) = [{ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q + (p + 2q − 1)2 [2]2p,q } [n + m]2p,q ([n + 1]p,q + β)2 − 4 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 2 ] x2 + [{ 1 + 2q [2]p,q + q2 − 1 [3]p,q + 2 (p + 2q − 1) [2]2p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + 4 α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 − 4 (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) − 4 α ([n + 1]p,q + β) ] bnx + [ (p2 x bn + 1 − x bn )n+mp,q [3]p,q + (p x bn + 1 − x bn )2n+2mp,q [2]2p,q + 4 α [2]p,q (p x bn + 1 − x bn )n+mp,q + 2α 2 ] b2n ([n + 1]p,q + β)2 , and βn,p,q(x) = ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x ) . 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Lett., 4 (2016), 1-8. 1Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technol- ogy, Ichchhanath Mahadev Dumas Road, Surat -395 007 (Gujarat), India 2L. 1627 Awadh Puri Colony Beniganj, Phase -III, Opp. Industrial Training Institute, Ayodhya Main Road, Faizabad-224 001, (Uttar Pradesh), India ∗Corresponding author: vishnunarayanmishra@gmail.com