EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 4, 2017, 890-907 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Stancu type generalization of modified Srivastava-Gupta operators Alok Kumar1, Vishnu Narayan Mishra2,3,∗, Dipti Tapiawala4,5 1Department of Computer Science, Dev Sanskriti Vishwavidyalaya, Haridwar, Uttarakhand, India 2 Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Madhya Pradesh, India 3L. 1627 Awadh Puri Colony Beniganj, Phase-III, Opposite-Industrial Training Institute (I.T.I.), Faizabad, Uttar Pradesh, India 4Department of Mathematics, C U Shah University, Gujarat, India 5AS and H Department (Mathematics), Sardar Vallabhbhai Patel Institute of Technology, Vasad, Gujarat, India Abstract. In this paper, we introduce a Stancu type generalization of modified Srivastava-Gupta operators. We obtain the moments of the operators and then prove the basic convergence theorem. Next, the Voronovskaja type asymptotic formula and some direct results for the above operators are discussed. Also, the rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied. Then, we obtain point-wise estimates using the Lipschitz type maximal function and two parameter Lipschitz-type space. Further, we study the A-statistical convergence of these operators. Lastly, we give better estimations of the above operators using King type approach. 2010 Mathematics Subject Classifications: 41A25, 26A15, 40A35. Key Words and Phrases: Srivastava-Gupta operators, rate of convergence, modulus of conti- nuity, weighted approximation, pointwise estimates, A-statistical convergence. 1. Introduction In the year 2003, Srivastava and Gupta [33] introduced a general family of summation- integral type operators {Gn,c} which includes some well-known operators as special cases. They obtained the rate of convergence for functions of bounded variation. For the details of special cases in [33], we refer the readers to [13], [20] and [31]. ∗Corresponding author. Email addresses: alokkpma@gmail.com (A. Kumar), vishnunarayanmishra@gmail.com (V. N. Mishra), tapiawalad@yahoo.com (D. Tapiawala) http://www.ejpam.com 890 c© 2017 EJPAM All rights reserved. A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 891 For f ∈ Cγ [0,∞) := {f ∈ C[0,∞) : |f(t)| ≤M(1+t)γ for some M > 0, γ > 0}, Srivastava and Gupta proposed a certain family of positive linear operators defined by Gn,c(f ;x) = n ∞∑ k=1 pn,k(x, c) ∫ ∞ 0 pn+c,k−1(t, c)f(t)dt+ pn,0(x, c)f(0), (1) where pn,k(x, c) = (−x)k k! φ(k)n,c(x) (2) and φn,c(x) = { e−nx, c = 0, (1 + cx)−n/c, c ∈ N . Verma and Agrawal [35] introduced the generalized form of the operators (1) and studied some of its approximation properties. Deo [3] gave a modification of these operators and established the rate of convergence and Voronovskaja type asymptotic result. Recently, Acar et al. [1] introduced Stancu type generalization of the operators (1) and obtained an estimate of the rate of convergence for functions having derivatives of bounded variation and also studied the simultaneous approximation for these operators. Yadav [36] proposed the modification of the operators (1) using the King approach as G∗n,c(f ;x) = n ∞∑ k=1 pn,k(x, c) ∫ ∞ 0 pn+c,k−1(t, c)f ( (n− c)t n ) dt+ pn,0(x, c)f(0) (3) and studied its moment estimates, direct estimate, asymptotic formula and statistical convergence. Recently, Maheshwari [22] obtained the rate of convergence for the functions having bounded derivatives on every finite subinterval of [0,∞) for the operators (3). Very recently, Neer et al. [29] introduced the Bezier variant of the operators (3) and studied the direct approximation result and estimate of the rate of convergence of these operators for functions of bounded variation. In [34], Stancu introduced the positive linear operators P (α,β) n : C[0, 1] → C[0, 1] by modifying the Bernstein polynomial as P (α,β) n (f ;x) = n∑ k=0 bn,k(x)f ( k + α n+ β ) , where bn,k(x) = ( n k ) xk(1 − x)n−k, x ∈ [0, 1] is the Bernstein basis function and α, β are any two real numbers which satisfy the condition that 0 ≤ α ≤ β. In the recent years, Stancu type generalization of the certain operators introduced by several researchers and obtained different type of approximation properties of many oper- ators, we refer some of the important papers in this direction as [1], [2], [32] etc. A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 892 For f ∈ Cγ [0,∞), 0 ≤ α ≤ β we introduce the following Stancu type generalization of the operators (3): G∗(α,β)n,c (f ;x) = n ∞∑ k=1 pn,k(x, c) ∫ ∞ 0 pn+c,k−1(t, c)f ( (n− c)t+ α n+ β ) dt +pn,0(x, c)f ( α n+ β ) (4) For α = β = 0, we denote G ∗(α,β) n,c (f ;x) by G∗n,c(f ;x). The goal of the present paper is to study the basic convergence theorem, Voronovskaja type asymptotic result, local approximation theorem, rate of convergence, weighted approxi- mation, pointwise estimation and A-statistical convergence of the operators (4). Further, to obtain better approximation, we also propose modification of the operators (4) using King type approach. 2. Moment Estimates Lemma 1. [36] For G∗n,c(t m;x), m = 0, 1, 2, one has (i) G∗n,c(1;x) = 1; (ii) G∗n,c(t;x) = x; (iii) G∗n,c(t 2;x) = (n2−c2)x2+2x(n−c) n(n−2c) , for n > 2c. Lemma 2. For the operators G ∗(α,β) n,c (f ;x) as defined in (4), the following equalities hold: (i) G ∗(α,β) n,c (1;x) = 1; (ii) G ∗(α,β) n,c (t;x) = nx+α n+β ; (iii) G ∗(α,β) n,c (t2;x) = { n(n2−c2) (n−2c)(n+β)2 } x2 + { 2n((n−c)+α(n−2c)) (n−2c)(n+β)2 } x+ α2 (n+β)2 , for n > 2c. Proof. For x ∈ [0,∞), in view of Lemma 1, we have G∗(α,β)n,c (1;x) = 1. Next, for f(t) = t, again applying Lemma 1, we get G∗(α,β)n,c (t;x) = n ∞∑ k=1 pn,k(x, c) ∫ ∞ 0 pn+c,k−1(t, c) ( (n− c)t+ α n+ β ) dt+ pn,0(x, c) ( α n+ β ) = n n+ β G∗n,c(t, x) + α n+ β = nx+ α n+ β A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 893 Proceeding similarly, we have G∗(α,β)n,c (t2;x) = n ∞∑ k=1 pn,k(x, c) ∫ ∞ 0 pn+c,k−1(t, c) ( (n− c)t+ α n+ β )2 dt+ pn,0(x, c) ( α n+ β )2 = ( n n+ β )2 G∗n,c(t 2, x) + 2nα (n+ β)2 G∗n,c(t, x) + ( α n+ β )2 = { n(n2 − c2) (n− 2c)(n+ β)2 } x2 + { 2n((n− c) + α(n− 2c)) (n− 2c)(n+ β)2 } x+ α2 (n+ β)2 . Lemma 3. For f ∈ CB[0,∞) (space of all bounded and continuous functions on [0,∞) endowed with norm ‖ f ‖= sup{|f(x)| : x ∈ [0,∞)}), ‖ G∗(α,β)n,c (f ;x) ‖≤‖ f ‖. Proof. In view of (4) and Lemma 2, the proof of this lemma easily follows. Remark 1. For every x ≥ 0, n > 2c, we have G∗(α,β)n,c ((t− x);x) = α− βx n+ β , and G∗(α,β)n,c ( (t− x)2;x ) = { nc(2n− c) + β2(n− 2c) (n− 2c)(n+ β)2 } x2 + { 2n(n− c)− 2αβ(n− 2c) (n− 2c)(n+ β)2 } x+ α2 (n+ β)2 , n > 2c = γ(α,β)n,c (x), (say). 3. Main Results Theorem 4. (Voronovskaja type theorem) Let f ∈ CB[0,∞). If f ′, f ′′ exists at a fixed point x ∈ [0,∞), we have lim n→∞ n ( G∗(α,β)n,c (f ;x)− f(x) ) = (α− βx)f ′(x) + x(1 + cx)f ′′(x). Proof. Let x ∈ [0,∞) be fixed. From the Taylor’s theorem, we may write f(t) = f(x) + (t− x)f ′(x) + 1 2 f ′′(x)(t− x)2 + ξ(t, x)(t− x)2, (5) where ξ(t, x) is the peano form of the remainder and lim t→x ξ(t, x) = 0. Applying G ∗(α,β) n,c (f, x) on both sides of (5), we have n ( G∗(α,β)n,c (f ;x)− f(x) ) = nf ′(x)G∗(α,β)n,c ((t− x);x) + 1 2 nf ′′(x)G∗(α,β)n,c ( (t− x)2;x ) A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 894 +nG∗(α,β)n,c ( (t− x)2ξ(t, x);x ) . In view of Remark 1, we have lim n→∞ nG∗(α,β)n,c ((t− x);x) = α− βx (6) and lim n→∞ nG∗(α,β)n,c ( (t− x)2;x ) = 2x(1 + cx). (7) Now, we shall show that lim n→∞ nG∗(α,β)n,c ( ξ(t, x)(t− x)2;x ) = 0 By using Cauchy-Schwarz inequality, we have G∗(α,β)n,c ( ξ(t, x)(t− x)2;x ) ≤ √ G ∗(α,β) n,c (ξ2(t, x);x) √ G ∗(α,β) n,c ((t− x)4;x). (8) We observe that ξ2(x, x) = 0 and ξ2(., x) ∈ CB[0,∞). Then, it follows that lim n→∞ G∗(α,β)n,c (ξ2(t, x);x) = ξ2(x, x) = 0, (9) in view of fact that G ∗(α,β) n,c ((t− x)4;x) = O ( 1 n2 ) . Now, from (8) and (9) we obtain lim n→∞ nG∗(α,β)n,c ( ξ(t, x)(t− x)2;x ) = 0. (10) From (6), (7) and (10), we get the required result. 3.1. Local approximation For CB[0,∞), let us consider the following K-functional: K2(f, δ) = inf g∈W 2 {‖ f − g ‖ +δ ‖ g′′ ‖}, where δ > 0 and W 2 = {g ∈ CB[0,∞) : g′, g ′′ ∈ CB[0,∞)}. By, p. 177, Theorem 2.4 in [4], there exists an absolute constant C > 0 such that K2(f, δ) ≤ Cω2(f, √ δ), (11) where ω2(f, √ δ) = sup 0 0, and defined as ωb(f, δ) = sup |t−x|≤δ sup x,t∈[0,b] |f(t)− f(x)|. We observe that for a function f ∈ CB[0,∞), the modulus of continuity ωb(f, δ) tends to zero. Now, we give a rate of convergence theorem for the operators G ∗(α,β) n,c . Theorem 6. Let f ∈ CB[0,∞) and ωb+1(f, δ) be its modulus of continuity on the finite interval [0, b+ 1] ⊂ [0,∞), where b > 0. Then, for every n > 2c, |G∗(α,β)n,c (f ;x)− f(x)| ≤ 4Mf (1 + b2)γ(α,β)n,c (x) + 2ωb+1 ( f, √ γ (α,β) n,c (x) ) , where γ (α,β) n,c (x) is defined in Remark 1 and Mf is a constant depending only on f. A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 897 Proof. For x ∈ [0, b] and t > b+ 1. Since t− x > 1, we have |f(t)− f(x)| ≤Mf (2 + t2 + x2) ≤Mf (t− x)2(2 + 2x+ 2x2) ≤ 4Mf (1 + b2)(t− x)2. For x ∈ [0, b] and t ≤ b+ 1, we have |f(t)− f(x)| ≤ ωb+1(f, |t− x|) ≤ ( 1 + |t− x| δ ) ωb+1(f, δ), δ > 0. From the above, we have |f(t)− f(x)| ≤ 4Mf (1 + b2)(t− x)2 + ( 1 + |t− x| δ ) ωb+1(f, δ), δ > 0. Thus, by applying Cauchy-Schwarz inequality, we have |G∗(α,β)n,c (f ;x)− f(x)| ≤ 4Mf (1 + b2)(G∗(α,β)n,c (t− x)2;x) +ωb+1(f, δ) ( 1 + 1 δ (G∗(α,β)n,c (t− x)2;x) 1 2 ) ≤ 4Mf (1 + b2)γ(α,β)n,c (x) + 2ωb+1 ( f, √ γ (α,β) n,c (x) ) , on choosing δ = √ γ (α,β) n,c (x). This completes the proof of the theorem. 3.3. Weighted approximation. Let Cν be the space of all continuous functions on [0,∞) with the norm ‖ f ‖ν= sup x∈[0,∞) |f(x)| ν(x) and C0 ν = {f ∈ Cν : lim x→∞ |f(x)| ν(x) <∞}, where ν(x) is a weight function. In what follows we consider ν(x) = 1 + x2. Theorem 7. For each f ∈ C0 ν , we have lim n→∞ ‖ G∗(α,β)n,c (f)− f ‖ν= 0. Proof. From [8], we know that it is sufficient to verify the following three conditions lim n→∞ ‖ G∗(α,β)n,c (tk;x)− xk ‖ν= 0, k = 0, 1, 2. (16) Since G ∗(α,β) n,c (1;x) = 1, the condition in (16) holds for k = 0. By Lemma 2, we have ‖ G∗(α,β)n,c (t;x)− x) ‖ν = sup x∈[0,∞) |G∗(α,β)n,c (t;x)− x| 1 + x2 A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 898 ≤ β n+ β sup x∈[0,∞) x 1 + x2 + α n+ β sup x∈[0,∞) 1 1 + x2 ≤ α+ β n+ β , which implies that the condition in (16) holds for k = 1. Similarly, we can write for n > 2c ‖ G∗(α,β)n,c (t2;x)− x2 ‖ν = sup x∈[0,∞) |G∗(α,β)n,c (t2;x)− x2| 1 + x2 ≤ ∣∣∣∣ n(n2 − c2) (n− 2c)(n+ β)2 − 1 ∣∣∣∣+ ∣∣∣∣2n((n− c) + α(n− 2c)) (n− 2c)(n+ β)2 ∣∣∣∣ + α2 (n+ β)2 , which implies that lim n→∞ ‖ G∗(α,β)n,c (t2;x)− x2 ‖ν= 0, the equation (16) holds for k = 2. This completes the proof of theorem. Now we give the following theorem to approximate all functions in C0 ν . Such type of results are given in [9] for locally integrable functions. Theorem 8. For each f ∈ C0 ν and σ > 0, we have lim n→∞ sup x∈[0,∞) |G∗(α,β)n,c (f ;x)− f(x)| (1 + x2)σ+1 = 0. Proof. For any fixed x0 > 0, sup x∈[0,∞) |G∗(α,β)n,c (f ;x)− f(x)| (1 + x2)σ+1 = sup x≤x0 |G∗(α,β)n,c (f ;x)− f(x)| (1 + x2)σ+1 + sup x>x0 |G∗(α,β)n,c (f ;x)− f(x)| (1 + x2)σ+1 sup x∈[0,∞) |G∗(α,β)n,c (f ;x)− f(x)| (1 + x2)σ+1 ≤ ‖ G∗(α,β)n,c (f)− f ‖C[0,x0] + ‖ f ‖ν sup x>x0 |G∗(α,β)n,c (1 + t2;x)| (1 + x2)σ+1 + sup x>x0 |f(x)| (1 + x2)σ+1 . The first term of the above inequality tends to zero from Theorem 6. By Lemma 2, for any fixed x0 > 0, it is easily prove that sup x>x0 |G∗(α,β)n,c (1 + t2;x)| (1 + x2)σ+1 → 0 as n → ∞. We can choose x0 > 0 so large that the last part of the above inequality can be small. Hence the proof is completed. A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 899 3.4. Pointwise Estimates In this section, we establish some pointwise estimates of the rate of convergence of the operators G ∗(α,β) n,c . First, we give the relationship between the local smoothness of f and local approximation. We know that a function f ∈ C[0,∞) is in LipM (α) on E, α ∈ (0, 1], E⊂ [0,∞) if it satisfies the condition |f(t)− f(x)| ≤M |t− x|α, t ∈ [0,∞) and x ∈ E, where M is a constant depending only on α and f . Theorem 9. Let f ∈ C[0,∞) ∩ LipM (α), E ⊂ [0,∞) and α ∈ (0, 1]. Then, we have |G∗(α,β)n,c (f ;x)− f(x)| ≤ M (( γ(α,β)n,c (x) )α/2 + 2dα(x,E) ) , x ∈ [0,∞), where M is a constant depending on α and f and d(x,E) is the distance between x and E defined as d(x,E) = inf{|t− x| : t ∈ E}. Proof. Let E be the closure of E in [0,∞). Then, there exists at least one point x0 ∈ E such that d(x,E) = |x− x0|. By our hypothesis and the monotonicity of G ∗(α,β) n,c , we get |G∗(α,β)n,c (f ;x)− f(x)| ≤ G∗(α,β)n,c (|f(t)− f(x0)|;x) +G∗(α,β)n,c (|f(x)− f(x0)|;x) ≤ M ( G∗(α,β)n,c (|t− x0|α;x) + |x− x0|α ) ≤ M ( G∗(α,β)n,c (|t− x|α;x) + 2|x− x0|α ) . Now, applying Hölder’s inequality with p = 2 α and 1 q = 1− 1 p , we obtain |G∗(α,β)n,c ((f ;x)− f(x)| ≤M ( {G∗(α,β)n,c (|t− x|2;x)}α/2 + 2dα(x,E) ) , from which the desired result immediate. Next, we obtain the local direct estimate of the operators defined in (4), using the Lipschitz-type maximal function of order α introduced by B. Lenze [19] as ω̃α(f, x) = sup t6=x, t∈[0,∞) |f(t)− f(x)| |t− x|α , x ∈ [0,∞) and α ∈ (0, 1]. (17) A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 900 Theorem 10. Let f ∈ CB[0,∞) and 0 < α ≤ 1. Then, for all x ∈ [0,∞) we have |G∗(α,β)n,c (f ;x)− f(x)| ≤ ω̃α(f, x) ( γ(α,β)n,c (x) )α/2 . Proof. From the equation (17), we have |G∗(α,β)n,c (f ;x)− f(x)| ≤ ω̃α(f, x)G∗(α,β)n,c (|t− x|α;x). Applying the Hölder’s inequality with p = 2 α and 1 q = 1− 1 p , we get |G∗(α,β)n,c (f ;x)− f(x)| ≤ ω̃α(f, x)G∗(α,β)n,c ((t− x)2;x) α 2 ≤ ω̃α(f, x) ( γ(α,β)n,c (x) )α/2 . Thus, the proof is completed. For a, b > 0, Özarslan and Aktuğlu [30] consider the Lipschitz-type space with two parameters: Lip (a,b) M (α) = ( f ∈ C[0,∞) : |f(t)− f(x)| ≤M |t− x|α (t+ ax2 + bx)α/2 ; x, t ∈ [0,∞) ) , where M is any positive constant and 0 < α ≤ 1. Theorem 11. For f ∈ Lip(a,b)M (α). Then, for all x > 0, we have |G∗(α,β)n,c (f ;x)− f(x)| ≤M ( γ (α,β) n,c (x) ax2 + bx )α/2 . Proof. First we prove the theorem for α = 1. Then, for f ∈ Lip(a,b)M (1), and x ∈ [0,∞), we have |G∗(α,β)n,c (f ;x)− f(x)| ≤ G∗(α,β)n,c (|f(t)− f(x)|;x) ≤ MG∗(α,β)n,c ( |t− x| (t+ ax2 + bx)1/2 ;x ) ≤ M (ax2 + bx)1/2 G∗(α,β)n,c (|t− x|;x). Applying Cauchy-Schwarz inequality, we get |G∗(α,β)n,c (f ;x)− f(x)| ≤ M (ax2 + bx)1/2 ( G∗(α,β)n,c ((t− x)2;x) )1/2 ≤ M ( γ (α,β) n,c (x) ax2 + bx )1/2 . A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 901 Thus the result holds for α = 1. Now, we prove that the result is true for 0 < α < 1. Then, for f ∈ Lip (a,b) M (α), and x ∈ [0,∞), we get |G∗(α,β)n,c (f ;x)− f(x)| ≤ M (ax2 + bx)α/2 G∗(α,β)n,c (|t− x|α;x). Taking p = 1 α and q = p p−1 , applying the Hölders inequality, we have |G∗(α,β)n,c (f ;x)− f(x)| ≤ M (ax2 + bx)α/2 ( G∗(α,β)n,c (|t− x|;x) )α . Finally by Cauchy-Schwarz inequality, we get |G∗(α,β)n,c (f ;x)− f(x)| ≤ M ( γ (α,β) n,c (x) ax2 + bx )α/2 . Thus, the proof is completed. 3.5. Statistical convergence Let A = (ank), (n, k ∈ N), be a non-negative infinite summability matrix. For a given sequence x := (x)n, the A-transform of x denoted by Ax : ((Ax)n) is defined as (Ax)n = ∞∑ k=1 ankxk provided the series converges for each n. A is said to be regular if lim n (Ax)n = L whenever lim n xn = L. The sequence x = (x)n is said to be a A- statistically convergent to L i.e. stA − lim n (x)n = L if for every ε > 0, lim n ∑ k:|xk−L|≥ε ank = 0. If we replace A by C1 then A is a Cesáro matrix of order one and A- statistical convergence is reduced to the statistical convergence. Similarly, if A = I, the identity matrix, then A- statistical convergence coincides with the ordinary convergence. It is to be noted that the concept of A-statistical convergence may also be given in normed spaces. Many researchers have investigated the statistical convergence properties for several sequences and classes of linear positive operators (see [5], [6], [7], [10], [23], [28]). In the following result we prove a weighted Korovkin theorem via A-statistical convergence. Throughout this section, let us assume that ei(t) = ti, i = 0, 1, 2. Theorem 12. Let (ank) be a non-negative regular infinite summability matrix and x ∈ [0,∞). Let νς ≥ 1 be a continuous function such that lim x→∞ ν(x) νς(x) = 0. A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 902 Then, for all f ∈ C0 ν , we have stA − lim n ‖ G∗(α,β)n,c (f)− f ‖νς= 0. Proof. From ([7] p. 195, Th. 6), it is enough to show that stA − lim n ‖ G∗(α,β)n,c (ei)− ei ‖ν= 0. From Lemma 2, we get stA − lim n ‖ G∗(α,β)n,c (e0)− e0 ‖ν= 0. Again by using Lemma 2, we have ‖ G∗(α,β)n,c (e1)− e1 ‖ν ≤ β (n+ β) sup x∈[0,∞) x 1 + x2 + α n+ β sup x∈[0,∞) 1 1 + x2 ≤ α+ β n+ β . For any given ε > 0, let us define the following sets: S := { n :‖ G∗(α,β)n,c (e1)− e1 ‖ν≥ ε } , S1 := { n : α n+ β ≥ ε 2 } and S2 := { n : β n+ β ≥ ε 2 } . Then, we get S ⊆ S1 ∪ S2 which implies that∑ k∈S ank ≤ ∑ k∈S1 ank + ∑ k∈S2 ank and hence stA − lim n ‖ G∗(α,β)n,c (e1)− e1 ‖ν= 0. Similarly, we have ‖ G∗(α,β)n,c (e2)− e2 ‖ν ≤ ( n(n2 − c2) (n− 2c)(n+ β)2 − 1 ) + 2n((n− c) + α(n− 2c)) (n− 2c)(n+ β)2 + α2 (n+ β)2 . Now, we define the following sets: U := { n :‖ G∗(α,β)n,c (e2)− e2 ‖ν≥ ε } , A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 903 U1 := { n : ( n(n2 − c2) (n− 2c)(n+ β)2 − 1 ) ≥ ε 3 } , U2 := { n : 2n((n− c) + α(n− 2c)) (n− 2c)(n+ β)2 ≥ ε 3 } and U3 := { n : α2 (n+ β)2 ≥ ε 3 } . Then, we get U ⊆ U1 ∪ U2 ∪ U3 which implies that∑ k∈U ank ≤ ∑ k∈U1 ank + ∑ k∈U2 ank + ∑ k∈U3 ank and hence stA − lim n ‖ G∗(α,β)n,c (e2)− e2 ‖ν= 0. This completes the proof of the theorem. 4. Better Estimates It is well known that the classical Bernstein polynomial preserve constant as well as linear functions. To make the convergence faster, King [18] proposed an approach to modify the Bernstein polynomial, so that the sequence preserve test functions e0 and e2, where ei(t) = ti, i = 0, 1, 2. As the operator G ∗(α,β) n,c (f ;x) defined in (4) preserve only the constant functions so further modification of these operators is proposed to be made so that the modified operators preserve the constant as well as linear functions. For this purpose the modification of (4) is defined as G ∗(α,β) n,c (f ;x) = n ∞∑ k=1 pn,k(rn(x), c) ∫ ∞ 0 pn+c,k−1(t, c)f ( (n− c)t+ α n+ β ) dt +pn,0(rn(x), c)f ( α n+ β ) , (18) where rn(x) = (n+β)x−α n for x ∈ In = [ α n+β ,∞) and n > 2c. Lemma 13. For each x ∈ In, by simple computations, we have (i) G ∗(α,β) n,c (1;x) = 1; (ii) G ∗(α,β) n,c (t;x) = x; A. Kumar, V. N. Mishra, D. Tapiawala / Eur. J. Pure Appl. Math, 10 (4) (2017), 890-907 904 (iii) G ∗(α,β) n,c (t2;x) = (n2 − c2) n(n− 2c) x2+ 2n(n− c)− 2αc(2n− c) n(n− 2c)(n+ β) x+ α2c(2n− c)− 2αn(n− c) n(n− 2c)(n+ β)2 . Consequently, for each x ∈ In , we have the following equalities G ∗(α,β) n,c (t− x;x) = 0 G ∗(α,β) n,c ((t− x)2;x) = c(2n− c) n(n− 2c) x2 + 2n(n− c)− 2cα(2n− c) n(n− 2c)(n+ β) x + α2c(2n− c)− 2αn(n− c) n(n− 2c)(n+ β)2 = ζ(α,β)n,c (x), (say). (19) Theorem 14. Let f ∈ CB(In) and x ∈ In. Then for n > 2c, there exists a positive constant C ′ such that |G∗(α,β)n,c (f ;x)− f(x)| ≤ C ′ω2 ( f, √ ζ (α,β) n,c (x) ) , where ζ (α,β) n,c (x) is given by (19). Proof. Let g ∈W 2 and x, t ∈ In. Using the Taylor’s expansion we have g(t) = g(x) + (t− x)g′(x) + ∫ t x (t− v)g′′(v)dv. Applying G ∗(α,β) n,c on both sides and using Lemma 13, we get G ∗(α,β) n,c (g;x)− g(x) = G ∗(α,β) n,c (∫ t x (t− v)g′′(v)dv;x ) . Obviously, we have ∣∣∣∣∫ t x (t− v)g′′(v)dv ∣∣∣∣ ≤ (t− x)2‖g′′‖. Therefore | G∗(α,β)n,c (g;x)− g(x) |≤ G∗(α,β)n,c ((t− x)2;x) ‖ g′′ ‖= ζ(α,β)n,c (x) ‖ g′′ ‖ . 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