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R E S E A R C H

Open Access

(

p

,

q

)

-gamma operators which preserve

x

2

Wen-Tao Cheng

1*

, Wen-Hui Zhang

2

and Qing-Bo Cai

3

*Correspondence:

chengwentao_0517@163.com 1School of Mathematics and

Computation Science, Anqing Normal University, Anhui, China Full list of author information is available at the end of the article

Abstract

In this paper, we introduce (p,q)-gamma operators which preservex2, we estimate the moments of these operators, and establish direct and local approximation theorems of these operators. Then two approximation theorems about Lipschitz functions are obtained. The estimates on the rate of convergence and some weighted approximation theorems of the operators are also obtained. Furthermore, the Voronovskaja-type asymptotic formula is also presented.

MSC: 41A10; 41A25; 41A36

Keywords: (p,q)-integers; (p,q)-gamma operators; Modulus of continuity; Rate of convergence

1 Introduction

With the rapid development of the approximation theory about the operators since the last century, lots of operators, such as Bernstein operators [4], Szász–Mirakjan operators [32,

37], Baskakov operators [3], Bleimann–Butzer–Hann operators [5], and Meyer–König– Zeller operators [31], have been proposed and constructed by several researchers due to Weierstrass and the important convergence theorem of Korovkin [26], see also [17]. In [23], Karsli considered gamma operators and studied the rate of convergence of these op-erators for the functions with derivative of bounded variation

Ln(f;x) =

(2n+ 3)!xn+3

n!(n+ 2)!

0

tn

(x+t)2n+4f(t) dt, x> 0. (1)

In [25], Karsli and Ozarslan established some local and global approximation results for the operatorsLn.

In recent years, with the rapid development ofq-calculus [22], the study of new poly-nomials and operators constructed withq-integer has attracted more and more attention. Lupas first introducedq-Bernstein polynomials [27], and Phillips [36] proposed otherq -analogue of Bernstein polynomials. Later, many researchers have performed studies in this field, and theq-analogue of classical operators and modified operators, such asq-Szász– Mirakjan operators [28], q-Baskakov operators [13], q-Meyer–König–Zeller operators [12], q-Bleimann–Butzer–Hann operators [11] q-Phillips operators [29], q-Baskakov– Kantorovich operators [20],q-Baskakov–Durrmeyer operators [19],q-Szász-beta oper-ators [18], andq-Meyer–König–Zeller–Durrmeyer operators [15], has been constructed;

(2)

see also [2]. In [6], Cai and Zeng definedq-gamma operators

Gn,q(f;x) =

[2n+ 3]!(qn+32x)n+3qn(n2+1)

[n]q![n+ 2]q!

0

tn

(qn+32x+t)2n+4

q

f(t) dqt, x> 0 (2)

and gave their approximation properties.

Then many operators have been constructed with two parameters (p,q)-integer based on post-quantum calculus ((p,q)-calculus) which has been used efficiently in many areas of sciences such as Lie group, different equations, hypergeometric series, physical sci-ences, and so on. Recently, approximation by sequences of linear positive operators has been transferred to operators with (p,q)-integer. Let us review some useful notations and definitions about (p,q)-calculus in [2,17,21].

Let 0 <q<p≤1. For each nonnegative integern, the (p,q)-integer [n]p,q, (p,q)-factorial

[n]p,q! are defined by

[n]p,q= pnqn

pq , n= 0, 1, 2, . . .

and

[n]p,q! =

⎧ ⎨ ⎩

[1]p,q[2]p,q· · ·[n]p,q, n≥1;

1, n= 0.

Further, the (p,q)-power basis is defined by

(xy)np,q= (x+y)(px+qy)p2x+q2y· · ·pn–1x+qn–1y

and

(xy)np,q= (xy)(pxqy)p2xq2y· · ·pn–1xqn–1y.

Letnbe a non-negative integer, the (p,q)-gamma function is defined as

Γp,q(n+ 1) =

(pq)n p,q

(pq)n = [n]p,q!, 0 <q<p≤1.

Aral and Gupta [1] proposed a (p,q)-beta function of the second kind form,n∈Nas follows:

Bp,q(m,n) =

0

xm–1

(1⊕px)m+n p,q

dp,qx

and gave the relation of the (p,q)-analogues of beta and gamma functions:

Bp,q(m,n) =

qΓp,q(m)Γp,q(n)

(pm+1qm–1)m2Γp,q(m+n)

(3)

As a special case, ifp=q= 1,B(m,n) =ΓΓ(m(m)Γ+n(n)). It is obvious that order is important for (p,q)-setting, which is the reason why a (p,q)-variant of beta function does not satisfy commutativity property, i.e.,Bp,q(m,n)=Bp,q(n,m).

LetCB[0,∞) be the space of all real-valued continuous bounded functionsf on the

in-terval [0,∞) endowed with the norm

f = sup x∈[0,∞)

f(x).

Letδ> 0 andC2

B[0,∞) ={g:g,gCB[0,∞)}, the followingK-functional is defined:

K(f;δ) = inf gC2B[0,∞)

fg +δg.

Using DeVore–Lorentz theorem (see [10]), there exists a constantC> 0 such that

K(f;δ)≤2(f;

δ), (3)

where

ω2(f;δ) = sup 0<|t|≤δ

sup x∈[0,∞)

f(x+ 2t) – 2f(x+t) +f(x)

is the second order modulus of smoothness off. Also, byω(f;δ) we denote the usual mod-ulus of continuity offCB[0,∞) defined as

ω(f;δ) = sup

0<|t|≤δ

sup x∈[0,∞)

f(x+t) –f(x).

LetBx2[0,∞) denote the function space of all functionsf such that|f(x)| ≤Cf(1 +x2),

whereCf is a positive constant depending onf. ByCx2[0,∞) we denote the subspace of all

continuous functions in the function spaceBx2[0,∞). ByC0

x2[0,∞) we denote the subspace

of all functionsfCx2[0,∞) for whichlimx→∞|f(x)|

1+x2 is endowed with the norm

f x2= sup

x∈[0,∞)

|f(x)| 1 +x2.

Fora> 0, the modulus of continuity off on [0,a] is defined as follows:

ωa(f;δ) = sup |yx|<δ

sup

0≤x,ya

f(y) –f(x).

As is known, iff is not uniformly continuous on [0,∞), we cannot getω(f;δ)→0 as

δ →0. In [38], Yuksel and Ispir defined the weighted modulus of continuityΩ(f;δ) =

sup0<hδ,x0|f(x+h)–f(x)|

1+(x+h)2 whilefCx02[0,∞) and proved the properties of monotone

increas-ing about Ω(f;δ) asδ> 0 and the inequalityΩ(f;λδ)≤(1 +λ)Ω(f;δ) whileλ> 0 and

fCx02[0,∞).

LetfCB[0,∞),M> 0, andγ ∈(0, 1]. We recall thatf ∈LipM(γ) if the following

in-equality

(4)

is satisfied. LetF be a subset of the interval [0,∞), we define thatf ∈LipM(γ,F) if the following inequality

f(x) –f(y)≤M|xy|γ, xFandy∈[0,∞)

holds.

Recently, Mursaleen first applied (p,q)-calculus in approximation theory and introduced the (p,q)-analogue of Bernstein operators [33], (p,q)-Bernstein–Stancu operators [34], (p,q)-Bernstein–Schurer operators [35] and investigated their approximation properties. In addition, many well-known approximation operators with (p,q)-integer, such as (p,q )-Bernstein–Stancu–Schurer–Kantorovich operators [8], (p,q)-Szász–Baskakov operators [16], (p,q)-Baskakov-beta operators [30] have been introduced. All this achievement mo-tivates us to construct the (p,q)-analogue of the gamma operator (1), as we know that many researchers have studied approximation properties of the gamma operators and their modifications (see [7,9,24,39]). The rest of the paper is organized as follows. In Sect.2, we define the (p,q)-gamma operators and obtain the moments and the central moments of them. In Sect.3, we study the properties of the (p,q)-gamma operators about Lipschitz condition. Then some direct theorems about local approximation, rate of con-vergence, weighted approximation, and Voronovskaja-type approximation are obtained.

2 (p,q)-gamma operators and moments

We first define the analogue of gamma operators via (p,q)-calculus as follows.

Definition 2.1 For n∈N,x∈(0,∞)and0 <q<p≤1,the(p,q)-gamma operators can be defined as follows:

Gpn,q(f;x) =x

n+3(qn+32)n+3pn2+7 2n+72

Bp,q(n+ 1,n+ 3)

0

tn

((pq)n+32xt)2n+4

p,q

f(t) dp,qt.

OperatorsGpn,qare linear and positive. Forp= 1, they turn out to be theq-gamma

op-erators defined in (2). We will derive the momentsGpn,q(tk;x) and the central moments Gpn,q((tx)k;x) fork= 0, 1, 2, 3, 4.

Lemma 2.1 For x∈(0,∞), 0 <q<p≤1,and k= 0, 1, . . . ,n+ 2,we have

Gpn,qtk;x=x

k(pq)kk22[n+k]p,q![nk+ 2]p,q!

[n]p,q![n+ 2]p,q!

. (4)

Proof Using the properties of (p,q)-beta function and (p,q)-gamma function, we have

Gpn,qtk;x=x

n+3(qn+32)n+3pn2+72n+72

Bp,q(n+ 1,n+ 3)

0

tn+k

((pq)n+32xt)2n+4

p,q

dp,qt

=x

n+3(qn+32)n+3pn2+72n+72

Bp,q(n+ 1,n+ 3)

0

1 (pq)(2n+3)(n+2)x2n+4

× tn+k

(1⊕ pt

xqn+ 32pn+ 52

)2n+4

p,q

(5)

=x

n+3(qn+32)n+3pn2+72n+72

Bp,q(n+ 1,n+ 3)

0

(xqn+32pn+52)n+k+1

(pq)(2n+3)(n+2)x2n+4

×

( t xqn+ 32pn+ 52

)n+k

(1⊕ pt

xqn+ 32pn+ 52

)2n+4

p,q

dp,q

t

xqn+32pn+52

=x

kpkn+52kqkn+32kB

p,q(n+k+ 1,nk+ 3) Bp,q(n+ 1,n+ 3)

=x

k(pq)kk22[n+k]p,q![nk+ 2]p,q!

[n]p,q![n+ 2]p,q!

.

Lemma2.1is proved.

Lemma 2.2 For x∈(0,∞), 0 <q<p≤1,the following equalities hold:

1.Gpn,q(1;x) = 1;

2.Gpn,q(t;x) =

p q(1 –

pn+1

[n+2]p,q)x; 3.Gpn,q(t2;x) =x2;

4.Gpn,q(t3;x) = [n+3]p,qx

3

(pq)32[n]p,q ;

5.Gpn,q(t4;x) =

[n+3]p,q[n+4]p,qx4

(pq)4[n]

p,q[n–1]p,q for n> 1.

Proof The proof of this lemma is an immediate consequence of Lemma2.1. Hence the

details are omitted.

Lemma 2.3 Let n> 1and x∈(0,∞),then for0 <q<p≤1,we have the central moments

as follows:

1. A(x) :=Gnp,q(tx;x) = ((

p q– 1) –

p q

pn+1

[n+2]p,q)x;

2. B(x) :=Gnp,q((tx)2;x) = –2((

p q– 1) –

p q

pn+1

[n+2]p,q)x

2; 3. Gpn,q((tx)4;x) = ([n+2]p,q[n+3]p,q[n+4]p,q–4(pq)

5

2[n–1]p,q[n+2]p,q[n+3]p,q

(pq)4[n–1]

p,q[n]p,q[n+2]p,q +

–4(pq)92[n–1]p,q[n]p,q[n+1]p,q+7(pq)4[n–1]p,q[n]p,q[n+2]p,q

(pq)4[n–1]p,q[n]p,q[n+2]p,q )x4.

Proof BecauseGpn,q(tx;x) =Gnp,q(t;x) –x,Gpn,q((tx)2;x) =Gpn,q(t2;x) – 2xGpn,q(t;x) +x2,

andGpn,q((tx)4;x) =Gnp,q(t4;x) – 4xGpn,q(t3;x) + 6x2Gpn,q(t2;x) – 4x3Gnp,q(t;x) +x4, and from

Lemma2.2, we obtain Lemma2.3easily.

Lemma 2.4 The sequences(pn), (qn)satisfy0 <qn<pn≤1such that pn→1,qn→1and

pn

nα,qnnβ, [n]pn,qn→ ∞as n→ ∞,then

lim

n→∞[n– 1]pn,qnGpn

,qn

n (tx;x) = –

α+β

2 x; (5)

lim

n→∞[n– 1]pn,qnG

pn,qn

n

(tx)2;x= (α+β)x2; (6)

lim

n→∞[n– 1]pn,qnG

pn,qn

n

(6)

Proof Using

lim

n→∞[n– 1]pn,qn

pn qn – 1 – pn qn

pnn+1

[n+ 2]pn,qn

= lim

n→∞[n+ 2]pn,qn

pn qn – 1 – pn qn

pn+1

n

[n+ 2]pn,qn

= lim n→∞

pn+2

nqnn+2 pnqn

pn–√qnqnpn qn

pnn+1

=αβ

2 –α= –

α+β

2 ,

we get (5) and (6) easily. Letk=n– 2, we have

[n+ 2]pn,qn[n+ 3]pn,qn[n+ 4]pn,qn =q3n[k]pn,qn+p

k n[3]pn,qn

q4n[k]pn,qn+p

k n[4]pn,qn

q5n[k]pn,qn+p

k n[5]pn,qn

q12n[k]3pn,qn+pknqn7[5]pn,qn+q8n[4]pn,qn+q9n[3]pn,qn

[k]2pn,qn.

Similarly, we can obtain

[n– 1]pn,qn[n+ 2]pn,qn[n+ 3]pn,qnq7n[k]3pn,qn+p

k n

q3

n[4]pn,qn+q4n[3]pn,qn

[k]2

pn,qn, [n– 1]pn,qn[n]pn,qn[n+ 2]pn,qnq

4

n[k]3pn,qn+p

k n

q3n+qn[3]pn,qn

[k]2pn,qn, [n– 1]pn,qn[n]pn,qn[n+ 1]pn,qnq

3

n[k]3pn,qn+p

k n

q2n+qn[2]pn,qn

[k]2pn,qn.

By Lemma2.3, we can have

Gpn,qn

n

(tx)4;x

An+

1 [k]pn,qn

Bn

x4,

whereAn=q12n – 4p

5 2

nq

19 2

n – 4p

9 2

nq

15 2

n + 7p4nq8nand Bn=pkn

q7n[5]pn,qn+q

8

n[4]pn,qn+q

9

n[3]pn,qn– 4(pnqn)

5 2q3

n[4]pn,qn+q

4

n[3]pn,qn

– 4(pnqn)

9 2q2

n+qn[2]pn,qn

+ 7(pnqn)4

q3n+qn[3]pn,qn

.

SetP=√pn,Q=√qn, by

An=P24– 4P5Q19– 4P9Q15+ 7P8Q16 ∼P9– 4P5Q4– 4P9+ 7P8Q

= 3P5P4–Q4–Q4P5–Q5– 7P8(PQ)

= (PQ)

3P5

3

i=0

PiQ3–iQ4

4

i=0

PiQ4–i– 7P8

(7)

we easily obtain

[n– 1]pn,qnAn∼[n]pn,qn(PQ)

3P5

3

i=0

PiQ3–iQ4

4

i=0

PiQ4–i– 7P8

pnnqnn pnqn

pnqn

pn+√qn

3P5

3

i=0

PiQ3–iQ4

4

i=0

PiQ4–i– 7P8

ab

2 (3×4 – 5 – 7) = 0.

Similarly,Bn∼5 + 4 + 3 – 4×(4 + 3) – 4×(1 + 2) + 7×(1 + 3) = 0, we obtain (7).

3 Approximation properties of(p,q)-gamma operators

In this section, we research the approximation properties of (p,q)-gamma operators. The following two theorems show approximation properties about Lipschitz functions.

Theorem 3.1 Let0 <q<p≤1 and F be any bounded subset of the interval[0,∞).If

fCB[0,∞)∩LipM(γ,F),then,for all x∈(0,∞),we have

Gpn,q(f;x) –f(x)≤MB(x)

γ

2 + 2dγ(x;F),

where d(x;F)is the distance between x and F defined by d(x;F) =inf{|xy|:yF}.

Proof LetFbe the closure ofFin [0,∞). Using the properties of infimum, there is at least a pointy0∈Fsuch thatd(x;F) =|xy0|. By the triangle inequality, we can obtain

Gpn,q(f;x) –f(x)≤Gpn,qf(x) –f(t);x

Gpn,qf(x) –f(y0);x

+Gpn,qf(t) –f(y0);x

MGpn,q|ty0|γ;x

+Gpn,q|xy0|γ;x

MGpn,q|xt|γ;x+ 2dγ(x;F).

Choosingk1=γ2 andk2=

2

2–γ and using the well-known Hölder inequality, we have

Gpn,q(f;x) –f(x)≤MGnp,q|xt|k1γ;xk11Gp,q

n

1k2;xk12 + 2dγ(x;F)

MGpn,q(xt)2;x

γ

2 + 2dγ (x;F)

=MB(x)

γ

2 + 2dγ (x;F).

This completes the proof.

Theorem 3.2 Let0 <q<p≤1.Then,for all f ∈LipM(γ),we have

(8)

Proof Using the monotonicity of the operatorsGnp,q and the Hölder inequality, we can

obtain

Gp,q

n (f;x) –f(x)≤Gnp,qf(t) –f(x);x

MGp,q n

|tx|γ;x

=MGnp,q|tx|2

γ

2;xMGp,q n

(tx)2;x

γ

2 =MBγ2(x).

The third theorem is a direct local approximation theorem for the operatorsGpn,q(f;x).

Theorem 3.3 Let0 <q<p≤1,fCB[0,∞).Then,for every x∈(0,∞),there exists a positive constant C1such that

Gpn,q(f;x) –f(x)≤C1ω2

f;B(x) +A2(x)+ωf;A(x).

Proof Forx∈(0,∞), we consider new operatorsHnp,q(f;x) defined by Hnp,q(f;x) =Gpn,q(f;x) +f(x) –fA(x) +x.

Using the operator above and Lemma2.3, we have

Hnp,q(tx;x) =Gnp,q(tx;x) –A(x) = 0.

Letx,t∈(0,∞) andgCB2[0,∞). Using Taylor’s expansion, we can obtain

g(t) =g(x) +g(x)(tx) +

t

x

g(u)(tu) du.

Hence,

Hnp,q(g;x) –g(x)=g(x)Hnp,q(tx);x+Hnp,q

t

x

g(u)(tu) du;x

Hnp,q

t

x

g(u)(tu) du;x

Gpn,q

t

x

g(u)(tu) du;x

A(x)+x x

g(u)A(x) +xudu

Gnp,q

t

x

g(u)(tu) du;x

+

A(x)+x x

g(u)A(x) +xudu

B(x) +A2(x)g.

Using|Gpn,q(f;x)| ≤ f , we have

Gpn,q(f;x) –f(x)

=Hnp,q(f;x) +fA(x) +x– 2f(x)

Hnp,q(fg;x) – (fg)(x)+Hnp,q(g;x) –g(x)+fA(x) +xf(x)

(9)

Taking infimum over all gC2

B[0,∞) and using (3), we can obtain the desired

asser-tion.

The fourth theorem is a result about the rate of convergence for the operatorsGpn,q(f;x):

Theorem 3.4 Let fCx2[0,∞), 0 <q<p≤1,and a> 0,we have

Gpn,q(f;x) –f(x)C(0,a]≤4Cf

1 +a2B(a) + 2ωa+1

f;B(a).

Proof For allx∈(0,a] andt>a+ 1, we easily have (tx)2≥(ta)2≥1, therefore,

f(t) –f(x)≤f(t)+f(x)≤Cf

2 +x2+t2

=Cf

2 +x2+ (xtx)2≤Cf

2 + 3x2+ 2(xt)2

Cf

4 + 3x2(tx)2≤4Cf

1 +a2(tx)2,

(8)

and for allx∈(0,a],t∈(0,a+ 1], andδ> 0, we have

f(t) –f(x)≤ωa+1

f,|tx|≤

1 +|tx|

δ

ωa+1(f;δ). (9)

From (8) and (9), we get

f(t) –f(x)≤4Cf

1 +a2(tx)2+

1 +|tx|

δ

ωa+1(f;δ).

By Schwarz’s inequality and Lemma2.3, we have

Gpn,q(f;x) –f(x)

Gpn,qf(t) –f(x);x

≤4Cf

1 +a2Gnp,q(tx)2;x+Gpn,q

1 +|tx|

δ

;x

ωa+1(f;δ)

≤4Cf

1 +a2Gnp,q(tx)2;x+ωa+1(f;δ)

1 +1

δ

Gpn,q

(tx)2;x

≤4Cf

1 +a2B(x) +ωa+1(f;δ)

1 +1

δ

B(x)

≤4Cf

1 +a2B(a) +ω

a+1(f;δ)

1 +1

δ

B(a)

.

By takingδ=√B(a) and supremum over allx∈(0,a], we accomplish the proof of

Theo-rem3.4.

The following three results are theorems about weighted approximation for the opera-torsGpn,q(f;x).

Theorem 3.5 Let fC0x2[0,∞)and the sequences(pn), (qn)satisfy0 <qn<pn≤1such that pn

(10)

such that,for all n>N andν> 0,the inequality

sup x∈(0,∞)

|Gpn,qn

n (f;x) –f(x)|

(1 +x2)32+ν

≤4√2Ω

f; 1 [n– 1]pn,qn

(10)

holds.

Proof Fort> 0,x∈(0,∞) andδ> 0, by the definition and properties ofΩ(f;δ), we get

f(t) –f(x)≤1 +x+|xt|2Ωf;|tx|

≤21 +x21 + (tx)21 +|tx|

δ

Ω(f;δ).

Usingpn

n→1,qnn→1, [n]pn,qn→ ∞asn→ ∞and Lemma2.4, there exists a positive integerN∈N+such that, for alln>N,

Gpn,qn

n

(tx)2;x≤ 2(1 +x

2)

[n– 1]pn,qn

, (11)

Gpn,qn

n

(tx)4;x≤1. (12)

SinceGpn,qn

n is linear and positive, we have

Gpn,qn

n (f;x) –f(x)≤2

1 +x2Ω(f;δ)

1 +Gpn,qn

n

(tx)2;x

+Gpn,qn

n

1 + (tx)2|tx|

δ ;x

. (13)

To estimate the second term of (13), applying the Cauchy–Schwarz inequality and (x+

y)22(x2+y2), we have

Gpn,qn

n

1 + (tx)2|tx|

δ ;x

≤√2Gpn,qn

n

1 + (tx)4;x12

Gpn,qn

n

(tx)2

δ2 ;x

1 2

.

By (11) and (12),

Gpn,qn

n

1 + (tx)2|tx|

δ ;x

≤2

2(1 +x2)12 δ[n– 1]pn,qn

.

Takingδ=√ 1

[n–1]pn,qn, we can obtain

Gpn,qn

n (f;x) –f(x)≤4 √

21 +x232Ω

f; 1 [n– 1]pn,qn

.

The proof is completed.

Theorem 3.6 Let the sequences(pn), (qn)satisfy0 <qn<pn≤1such that pn→1,qn→1, and pn

nα,qnnβ, [n]pn,qn→ ∞as n→ ∞.Then,for fC

0

x2[0,∞),we have

lim n→∞G

pn,qn

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Proof By the Korovkin theorem in [14], we see that it is sufficient to verify the following three conditions:

lim n→∞G

pn,qn

n

tk;xxkx2= 0, k= 0, 1, 2. (15)

SinceGpn,qn

n (1;x) = 1,Gpnn,qn(t2;x) =x2, then (15) holds true fork= 0, 2. By Lemma2.2, we

can get

Gpn,qn

n (t;x) –xx2= sup

x∈(0,∞)

1 1 +x2G

pn,qn

n (t;x) –x

= sup x∈(0,∞)

x

1 +x2

pn–√qn

qn

pn qn

pn+1

n

[n+ 2]pn,qn

≤ sup

x∈(0,∞)

pn–√qn

qn

pn qn

pn+1

n

[n+ 2]pn,qn

→0, n→ ∞.

Thus the proof is completed.

Theorem 3.7 Let the sequences(pn), (qn)satisfy0 <qn<pn≤1such that pn→1,qn→1,

[n]pn,qn→ ∞as n→ ∞.For every fCx2[0,∞)andκ> 0,we have

lim n→∞xsup(0,)

|Gpn,qn

n (f;x) –f(x)|

(1 +x2)1+κ = 0.

Proof Letx0∈(0,∞) be arbitrary but fixed. Then

sup x∈(0,∞)

|Gpn,qn

n (f;x) –f(x)|

(1 +x2)1+κ ≤ sup

x∈(0,x0] |Gpn,qn

n (f;x) –f(x)|

(1 +x2)1+κ + sup

x∈(x0,∞) |Gpn,qn

n (f;x) –f(x)|

(1 +x2)1+κ

Gpn,qn

n (f;x) –f(x)C(0,x0]

+Cf sup x∈(x0,∞)

|Gpn,qn

n ((1 +t2);x)|

(1 +x2)1+κ + sup

x∈(x0,∞) |f(x)|

(1 +x2)1+κ. (16)

Since|f(x)| ≤Cf(1 +x2), we havesupx∈(x0,∞)

|f(x)|

(1+x2)1+κ

Cf

(1+x2 0)κ

. Let> 0 be arbitrary. We can choosex0to be so large that

Cf

(1 +x20)κ <. (17)

In view of Lemma2.2, whilex∈(x0,∞), we obtain

Cf lim n→∞

|Gpn,qn

n ((1 +t2);x)|

(1 +x2)1+κ =Cf

(1 +x2)

(1 +x2)1+κ =

Cf

(1 +x2)κ

Cf

(1 +x2 0)κ

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Using Theorem3.4, we can see that the first term of inequality (16) implies that

Gpn,qn

n (f;x) –f(x)C(0,x0]<, asn→ ∞. (18)

Combining (16)–(18), we get the desired result. The last result is a Voronovskaja-type asymptotic formula for the operatorsGpn,q(f;x).

Theorem 3.8 Let fCB2[0,∞)and the sequences(pn), (qn)satisfy0 <qn<pn≤1such that pn→1,qn→1and pnnα,qnnβ, [n]pn,qn→ ∞as n→ ∞,where0≤α,β< 1.

Then,for all x∈(0,∞),

lim

n→∞[n– 1]pn,qn

Gpn,qn

n (f;x) –f(x)

=α+β 2

xf(x) +x2f(x). (19)

Proof Letx∈(0,∞) be fixed. By Taylor’s expansion formula, we obtain

f(t) =f(x) +f(x)(tx) +

1 2f

(x) +Θ

pn,qn(t,x)

(tx)2,

where Θpn,qn(x,t) is bounded and limtxΘpn,qn(t,x) = 0. By applying the operator

Gpn,qn

n (f;x) to the relation above, we obtain

Gpn,qn

n (f;x) –f(x) =f(x)Gpnn,qn

(tx);x+1 2f

(x)Gpn,qn

n

(tx)2;x

+Gpn,qn

n

Θpn,qn(t,x)(tx)

2;x.

SincelimtxΘpn,qn(t,x) = 0, then for all> 0, there exists a positive constantδ> 0 which implies|Θpn,qn(t,x)|<for all fixedx∈(0,∞), wherenis large enough, while|tx| ≤δ, then|Θpn,qn(t,x)|<

C2

δ2(tx)2, whereC2is a positive constant. Using Lemma2.4, we obtain

[n– 1]pn,qnG

pn,qn

n

Θ(t,x)(tx)2;x

[n– 1]pn,qnG

pn,qn

n

(tx)2;x

+C2

δ2[n– 1]pn,qnG

pn,qn

n

(tx)4;x→0 (n→ ∞).

The proof is completed.

Acknowledgements

The authors are thankful to the editor and anonymous referees for their helpful comments and suggestions.

Funding

This research is supported by the National Natural Science Foundation of China (Grant No. 11626031 and Grant No. 11601266), the Philosophy and Social Sciences General Planning Project of Anhui Province of China (Grant No. AHSKYG2017D153), the Natural Science Foundation of Anhui Province of China (Grant No. 1908085QA29), the Natural Science Foundation of Fujian Province of China (Grant No. 2016J05017), the Natural Science Foundation of Anhui Province of China (Grant No. 1908085QA29), the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant No. 2018C087R), and the Program for New Century Excellent Talents in Fujian Province University. We also thank Fujian Provincial Key Laboratory of Data-Intensive Computing, Fujian University Laboratory of Intelligent Computing and Information Processing and Fujian Provincial Big Data Research Institute of Intelligent Manufacturing of China.

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Author details

1School of Mathematics and Computation Science, Anqing Normal University, Anhui, China.2Qingdao Hengxing

University of Science and Technology, Shangdong, China. 3School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou, China.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 26 December 2018 Accepted: 7 April 2019

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