Approximation by complex q-Durrmeyer polynomials in compact disks

Acta Mathematicae Applicatae Sinica, English Series

Vol. 30, No. 1 (2014) 65–74

DOI: 10.1007/s10255-014-0268-0http://www.ApplMath.com.cn & www.SpringerLink.com

Acta Mathema�cae Applicatae Sinica,English Series© The Editorial Office of AMAS & Springer-Verlag Berlin Heidelberg 2014

Approximation by Complex q-Durrmeyer Polynomialsin Compact DisksN.I. Mahmudov1, Vijay Gupta2

1Department of Mathematics, Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey

(Email: nazim.mahmudov@emu.edu.tr2School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka New Delhi-110078, India

(E-mail: vijaygupta2001@hotmail.com

Abstract In this paper, the order of approximation and Voronovskaja type results with quantitative estimate

for complex q-Durrmeyer polynomials attached to analytic functions on compact disks are obtained.

Keywords q-complex Durrmeyer polynomials; q-integers; q-derivatives; Voronovskaja-type asymptotic for-

mula

2000 MR Subject Classification 41A25; 30E10

1 Introduction

In the recent years the quantitative Voronovskaja-type results for various approximationoperators were studied by Gal and collaborators, for ready reference we mention the recentbook due to Sorin G. Gal[2]. Also upper quantitative estimates for the uniform convergencewere obtained for the first time, by Gal recently in his recent book although Durrmeyer typeoperators were not studied there. Very recently Mahmudov[5] established quantitative estimatesand asymptotic formula for genuine Bernstein polynomials. Anastassiou-Gal[1] established theapproximation properties of the complex Bernstein-Durrmeyer operator defined by

Dn(f, z) = (n + 1)n∑

k=0

pn,k(z)∫ 1

0

f(t)pn,k(t) dt, z ∈ C, (1)

where

pn,k(z) :=(

nk

)zk(1 − z)n−k.

In the last decade the applications of q calculus in approximation theory was started whenPhillips[6] considered the q Bernstein polynomials. Later many researchers studied approxima-tion properties for several other linear positive operators in real domain. Not much work onthe q operators is available in the literature, this is the main motivation of the present articleto extend the studies and here we study complex q Bernstein Durrmeyer operators. First wemention some basic notations of q calculus. For n ∈ N,

[n]q := 1 + q + q2 + · · · + qn−1,

[n]q! :={

[n]q[n − 1]q · · · [1]q, n = 1, 2, · · · ,1, n = 0.

Manuscript received April 12, 2011. Revised February 20, 2012.

66 N.I. MAHMUDOV, V. GUPTA

The q-binomial coefficients are given by[

nk

]

q

=[n]q!

[k]q![n − k]q!, 0 ≤ k ≤ n.

The q-derivative Dqf of a function f is defined by

(Dqf)(t) =f(t) − f(qt)

(1 − q)t, if t �= 0.

The q analogue of the Bernstein Durrmeyer operators was introduced by Gupta[3] and someapproximation properties in real case for these operators were also studied in [4]. In the complexcase for z ∈ C the q-Bernstein Durrmeyer operators for 0 < q < 1, can be defined as

Dn,q(f ; z) = [n + 1]qn∑

k=0

q−kpn,k(q; z)∫ 1

0

pn,k(q; qt)f(t)dqt, (2)

where

pn,k(q; z) =[

nk

]

q

zk(1 − z)n−kq , pn,n(q; z) = zn

with the q-Beta function given by

Bq(m, n) =[m − 1]q[n − 1]q

[m + n − 1]q=

∫ 1

0

tm−1(1 − qt)n−1q dqt

and

(1 − z)mq :=

{(1 − z)(1 − qz) · · · (1 − qm−1z), m = 1, 2, · · · ,1, m = 0.

The present paper deals with the approximation properties of the operators (2), we firstestimate moment estimates, recurrence formula for moments, the upper bound and finally wepresent a quantitative asymptotic formula.

2 Auxiliary Results

In the sequel, we need the following results:

Lemma 1. Dn,q(tm; z) is a polynomial of degree less than or equal to min(m, n) and

Dn,q(tm; z) =[n + 1]q!

[n + m + 1]q!

m∑

s=0

cs(m, q)[n]sqBn,q(ts; z).

where cs(m, q) ≥ 0 are constants depending on m and q and

Bn,q(f ; z) =n∑

k=0

[nk

]zk

n−k−1∏

s=0

(1 − qsz)f([k]q/[n]q

)

denotes the q-Bernstein polynomials.

Proof. By the definition of q-Beta function, we have

Dn,q(tm; z) =[n + 1]qn∑

k=0

q−kpn,k(q; z)∫ 1

0

pn,k(q; qt)tmdqt

=[n + 1]q!

[n + m + 1]q!

n∑

k=0

pn,k(q; z)[k + m]q!

[k]q!.

Summation-integral Type Operators 67

For m = 1, we have

Dn,q(t; z) =[n + 1]q![n + 2]q!

n∑

k=0

pn,k(q; z)[k + 1]q =1

[n + 2]q

n∑

k=0

pn,k(q; z)[n]q1 + q[k]q

[n]q

=1

[n + 2]q

1∑

s=0

[n]sqBn,q(ts; z),

thus the result is true for m = 1 with c0(1, q) = 1 > 0, c1(1, q) = q > 0.For m = 2, we have

Dn,q(t2; z) =[n + 1]q![n + 3]q!

n∑

k=0

pn,k(q; z)[k + 2]q[k + 1]q

=[n + 1]q![n + 3]q!

n∑

k=0

pn,k(q; z)(1 + q + (q + 2q2)[k]q + q3[k]2q)

=[n + 1]q![n + 3]q!

[(1 + q) + (q + 2q2)[n]qBn,q(t; z) + q3[n]2qBn,q(t2; z)]

=[n + 1]q![n + 3]q!

2∑

s=0

cs(2, q)[n]sqBn,q(ts; z),

thus the result is true for m = 2 with c0(2, q) = 1 + q > 0, c1(2, q) = q + 2q2 > 0 andc2(2, q) = q3 > 0. Continuing in this way, the result follows by mathematical induction. �

Also, the following lemma holds.

Lemma 2. For all m, n ∈ N the identity

[n + 1]q![n + m + 1]q!

m∑

s=0

cs(m, q)[n]sq ≤ 1

hold.

Proof. By Lemma 1,

Dn,q(em; 1) =[n + 1]q!

[n + m + 1]q!

m∑

s=0

cs(m, q)[n]sqBn,q(es; 1) =[n + 1]q!

[n + m + 1]q!

m∑

s=0

cs(m, q)[n]sq.

On the other hand,

Dn,q(em; 1) =[n + 1]qpn,n(q; 1)∫ 1

0

pn,n(q; qt)tmdqt

=[n + 1]q∫ 1

0

qntn+mdqt = qn[n + 1]q1

[n + m + 1]q≤ 1.

�

Lemma 2 implies that for all m, n ∈ N and |z| ≤ r we have(|Bn,q(es; z)| ≤ rs: it is known

68 N.I. MAHMUDOV, V. GUPTA

by Gal (see [2, p.61, Proof of Theorem 1.5.6] (last line)).

∣∣Dn,q(em; z)∣∣ ≤ [n + 1]q!

[n + m + 1]q!

m∑

s=0

cs(m, q)[n]sq |Bn,q(es; z)|

≤ [n + 1]q![n + m + 1]q!

m∑

s=0

cs(m, q)[n]sqrs

≤ [n + 1]q![n + m + 1]q!

m∑

s=0

cs(m, q)[n]sqrm ≤ rm, r ≥ 1.

Notice that if Pm(z) is a polynomial of degree m then by the Bernstein inequality and thecomplex mean value theorem we have

|DqPm(z)| ≤ ‖P ′m‖r ≤ m

r|Pm(z)| . (3)

Lemma 3. For all p ∈ N ∪ {0}, n ∈ N and z ∈ C, we have

Dn,q(ep+1; z) =z(1 − z)

q[n]q + q−p−1[p + 2]qDq(Dn,q(ep; z))

+[n]qz + q−p−1[p + 1]qq[n]q + q−p−1[p + 2]q

Dn,q(ep; z)

=qp+1z(1 − z)[n + p + 2]q

Dq(Dn,q(ep; z)) +qp+1[n]qz + [p + 1]q

[n + p + 2]qDn,q(ep; z).

Proof. By simple calculation we obtain

z(1 − z)Dq(pn,k(q; z)) = ([k]q − [n]qz)pn,k(q; z)

andt(1 − qt)Dq(pn,k(q; qt)) = t(1 − qt)([k]q − [n]qqt).

Using these identities, it follows that

z(1 − z)Dq(Dn,q(ep; z)) =[n + 1]qn∑

k=0

q−k([k]q − [n]qz)pn,k(q; z)∫ 1

0

pn,k(q; qt)tpdqt

=[n + 1]qn∑

k=0

q−kpn,k(q; z)∫ 1

0

([k]q − [n]qqt + [n]qqt)pn,k(q; qt)tpdqt

− z[n]q[n + 1]qn∑

k=0

q−kpn,k(q; z)∫ 1

0

pn,k(q; qt)tpdqt

=[n + 1]qn∑

k=0

q−kpn,k(q; z)∫ 1

0

(Dqpn,k(q; qt))t(1 − qt)tpdqt

+ [n]qqDn,q(ep+1; z) − z[n]qDn,q(ep; z). (4)

Using q-integration by parts, setting

δ(t) =t

q(1 − t)

( t

q

)p

=1

qp+1(tp+1 − tp+2),

Summation-integral Type Operators 69

the q-integral in the above formula becomes∫ 1

0

Dq(f(t))δ(qt)dqt = δ(t)f(t)∣∣01 −

∫ 1

0

f(t)Dqδ(t)dqt.

Thus∫ 1

0

Dq(pn,k(q; qt))t(1 − qt)tpdqt = δ(t)pn,k(q; qt)∣∣10−

∫ 1

0

pn,k(q; qt)Dqδ(t)dqt

= − q−p−1

∫ 1

0

pn,k(q; qt)Dq(tp+1 − tp+2)dqt = −q−p−1[p + 1]q∫ 1

0

pn,k(q; qt)tpdqt

+ q−p−1[p + 2]q∫ 1

0

pn,k(q; qt)tp+1dqt. (5)

Substituting (5) in (4), we get

z(1 − z)DqDn,q(ep; z) = − q−p−1[p + 1]qDn,q(ep; z) + q−p−1[p + 2]qDn,q(ep+1; z)+ [n]qqDn,q(ep+1; z) − z[n]qDn,q(ep; z)

=(q[n]q + q−p−1[p + 2]q)Dn,q(ep+1; z)− (z[n]q + q−p−1[p + 1]q)Dn,q(ep; z),

which implies the recurrence in the statement. �

3 Upper Estimation

In this section we present the following upper estimation:

Theorem 4. Let 1 ≤ r < R. Then for all |z| ≤ r we have

|Dn,q(f ; z)− f(z)| ≤ 1 + r

[n + 1]q

∞∑

p=1

|ap| (p + 1)(p + 2)rp−1.

Proof. From the recurrence formula

Dn,q(ep+1; z) =qp+1z(1 − z)[n + p + 2]q

Dq(Dn,q(ep; z)) +qp+1[n]qz + [p + 1]q

[n + p + 2]qDn,q(ep; z),

Dn,q(ep; z) − ep(z) =qpz(1 − z)[n + p + 1]q

Dq(Dn,q(ep−1; z)) +qp[n]qz + [p]q[n + p + 1]q

(Dn,q(ep−1; z)− ep−1(z))

+qp[n]qz + [p]q[n + p + 1]q

zp−1 − zp

=qpz(1 − z)[n + p + 1]q

Dq(Dn,q(ep−1; z)) +qp[n]qz + [p]q[n + p + 1]q

(Dn,q(ep−1; z)− ep−1(z))

+[p]q

[n + p + 1]qzp−1 − [p]q + qn+p

[n + p + 1]qzp

and the inequality (3) for p ≥ 1 we get

|Dn,q(ep; z) − ep(z)|≤r(1 + r)

[n + 1]qp − 1

r‖Dn,q(ep−1)‖r

70 N.I. MAHMUDOV, V. GUPTA

+ r |Dn,q(ep−1; z) − ep−1(z)| + [p + 1]q[n + 1]q

rp−1(1 + r)

≤ (1 + r)(p − 1)[n + 1]q

rp−1 + r |Dn,q(ep−1; z) − ep−1(z)| + [p + 1]q[n + 1]q

rp−1(1 + r)

≤2(p + 1)(1 + r)[n + 1]q

rp−1 + r |Dn,q(ep−1; z) − ep−1(z)| .

By writing the last inequality for p = 1, 2, · · · , we easily obtain, step by step the following

|Dn,q(ep; z) − ep(z)|≤r

(r |Dn,q(ep−2; z) − ep−2(z)| + 2p

[n + 1]q(1 + r)rp−2

)+ 2(p + 1)

(1 + r)[n + 1]q

rp−1

=r2 |Dn,q(ep−2; z) − ep−2(z)| + 2(1 + r)[n + 1]q

rp−1(p + 1 + p)

≤ · · · ≤ (1 + r)[n + 1]q

(p + 1)(p + 2)rp−1.

Next, we prove that Dn,q(f ; z) =∞∑

p=0apDn,q(ep, z). Indeed denoting fm(z) =

m∑j=0

ajzj,

|z| ≤ r with m ∈ N, by the linearity of Dn, we have

Dn,q(fm, z) =m∑

p=0

apDn,q(ep, z),

and it is sufficient to show that for any fixed n ∈ N and |z| ≤ r with r ≥ 1, we havelim

m→∞Dn,q(fm, z) = Dn,q(f ; z). But this is immediate from limm→∞ ||fm − f ||r = 0, the norm

being the defined as ||f ||r = max {|f(z)| : |z| ≤ r} and from the inequality

|Dn,q(fm; z) − Dn,q(f ; z)| ≤ [n + 1]qn∑

k=0

|pn,k(q; z)|q1−k

∫ 1

0

pn,k(q, qt)|fm(t) − f(t)|dqt

≤Cr,n||fm − f ||r,

valid for all |z| ≤ r, where

Cr,n = (n + 1)n∑

k=0

[nk

]

q

(1 + r)n−krk

∫ 1

0

pn,k(q; qt)dqt.

Therefore we get

|Dn,q(f ; z) − f(z)| ≤∞∑

p=0

|ap| · |Dn,q(ep, z) − ep(z)| =∞∑

p=1

|ap| · |Dn,q(ep, z) − ep(z)|,

as Dn,q(e0, z) = e0(z) = 1. It follows that

|Dn,q(f ; z) − f(z)| ≤∞∑

p=1

|ap| |Dn,q(ep; z)− ep(z)| ≤ 1 + r

[n + 1]q

∞∑

p=1

|ap| (p + 1)(p + 2)rp−1.

�

Summation-integral Type Operators 71

4 Voronovskaja-Type Result

The following Voronovskaja-type result with a quantitative estimate holds.

Theorem 5. Let 0 < q < 1, R > 1 and suppose that f : DR → C is analytic in DR = {z ∈C : |z| < R} that is we can write f(z) =

∞∑p=0

apzp, for all z ∈ DR. For any fixed r ∈ [1, R] and

for all n ∈ N, |z| ≤ r, we have∣∣∣∣Dn,q(f ; z)− f(z) − z(1 − z)f ′′(z) + (1 − 2z)f ′(z)

[n]q

∣∣∣∣ ≤Mr(f)[n]2q

+ (1 − q)∞∑

p=1

|cp|prp,

where Mr(f) =∞∑

p=1|ap|pFpr

p < ∞ and

Fp = p(p − 1)(p − 2) + 5p3 + 4p2(p + 1) + 4(p − 1)p(p + 1).

Proof. We denote ep(z) = zp, p = 0, 1, 2, · · · and πp,n(q; z) = Dn,q(ep; z). By the proof of

Theorem 4, we can write Dn,q(f ; z) =∞∑

p=0apπp,n(q; z). Also since

z(1 − z)f ′′(z) + (1 − 2z)f ′(z)[n]q

=z(1 − z)

[n]q

∞∑

p=2

app(p − 1)zp−2 +(1 − 2z)

[n]q

∞∑

p=1

appzp−1.

Thus∣∣∣Dn,q(f ; z) − f(z) − z(1 − z)f ′′(z) + (1 − 2z)f ′(z)

[n]q

∣∣∣

≤∞∑

p=1

|ap|∣∣∣πp,n(q; z) − ep(z) − (p2 − p(p + 1)z)zp−1

[n]q

∣∣∣,

for all z ∈ DR, n ∈ N.By Lemma 3, for all n ∈ N, z ∈ C and p = 0, 1, 2, · · ·, we have

Dn,q(ep+1; z) =qp+1z(1 − z)[n + p + 2]q

Dq(Dn,q(ep; z)) +qp+1[n]qz + [p + 1]q

[n + p + 2]qDn,q(ep; z).

If we denote

Ep,n(q; z) = Dn,q(ep; z) − ep(z) − (p(p − 1) − p2z)zp−1

[n]q,

then it is obvious that Ep,n(q; z) is a polynomial of degree less than or equal to p and by simplecomputation and the use of above recurrence relation, we are led to

Ep,n(q; z) =qpz(1 − z)[n + p + 1]q

Dq(Dn,q(ep−1; z) − ep−1(z))

+qp[n]qz + [p]q[n + p + 1]q

Ep−1,n(q; z) + Xp,n(q; z),

72 N.I. MAHMUDOV, V. GUPTA

where

Xp,n(q; z) =zp−2

[n]q[n + p + 1]q

[[p]q(p − 1)(p − 2) + z

(qp[n]q[p − 1]q

+ qp[n]q(p − 1)2 + [p]q[n]q − [p]qp(p − 1) − p2[n + p + 1]q)

+ z2(p(p + 1)[n + p + 1]q − [p]q[n]q − qn+p[n]q

− qp[n]q[p − 1]q − qp[n]qp(p − 1))]

=:zp−2

[n]q[n + p + 1]q(Ap,n(q) + zBp,n(q) + z2Cp,n(q)).

It is clear that|Ap,n(q)| ≤ p(p − 1)(p − 2).

On the other hand

Bp,n(q) =qp[n]q[p − 1]q+ qp[n]q(p − 1)2 + [p]q[n]q − [p]qp(p − 1) − p2[n + p + 1]q= [n]q(qp[p − 1]q + [p]q − qp(1 − 2p))− [p]q(p − 1)2 − p(p − 1)[p]q − p(p − 1)qn+p

and

[n]q(qp[p − 1]q + [p]q − qp(2p − 1)) = [n]q([2p − 1]q − qp(2p − 1))

=[n]q(1 − q)2p−1∑

j=1

[j]qq2p−1−j = (1 − qn)2p−1∑

j=1

[j]qq2p−1−j .

So|Bp,n(q)| ≤ (2p − 1)[2p − 1]q + (p − 1)2[p]q + p(p − 1)[p]q + p(p − 1) ≤ 5p3.

Now we estimate Cp,n(q):

Cp,n(q)=p(p + 1)[n + p + 1]q − [p]q[n]q − qn+p[n]q − qp[n]q[p − 1]q − qp[n]qp(p − 1)=qpp(p + 1)[n]q + p(p + 1)[p]q + qn+pp(p + 1)

− [p]q[n]q − qn+p[n]q − qp[n]q[p − 1]q − qp[n]qp(p − 1)=2qpp[n]q − [p]q[n]q − qn+p[n]q − qp[n]q[p − 1]q + p(p + 1)[p]q + qn+pp(p + 1)=[n]q(2qpp − [p]q − qp[p − 1]q) −−qn+p[n]q + p(p + 1)[p]q + qn+pp(p + 1)=[n]q(qp(2p − 1) − [2p− 1]q) − qp(qn − 1)[n]q + p(p + 1)[p]q + qn+pp(p + 1)

= − (1 − qn)2p−1∑

j=1

[j]qq2p−1−j + qp(1 − qn)[n]q + p(p + 1)[p]q + qn+pp(p + 1).

It follows that

|Cp,n(q)| ≤(2p − 1)[2p − 1]q + (1 − qn)[n]q + 2p2(p + 1)≤4(p + 1)p2 + (1 − qn)[n]q.

Thus

|Xp,n(q; z)| ≤rp−2

[n]2q(p(p − 1)(p − 2) + 5p3r + 4r2p2(p + 1)) +

rp

[n]q(1 − qn)

=rp−2

[n]2q(p(p − 1)(p − 2) + 5p3r + 4r2p2(p + 1)) + rp(1 − q)

Summation-integral Type Operators 73

for all p ≥ 1, n ∈ N and |z| ≤ r.Using the estimate in the proof of Theorem 4, we have

|Dn,q(ep; z) − ep(z)| ≤ (1 + r)[n + 1]q

(p + 1)(p + 2)rp−1

for all p, n ∈ N, |z| ≤ r, with 1 ≤ r. For all k, n ∈ N, p ≥ 1 and |z| ≤ r, it follows

|Ep,n(q; z)| ≤ qpr(1 + r)[n + p + 2]q

|E′p−1,n(q; z)| + qp+1[n]qz + [p + 1]q

[n + p + 2]q|Ep−1,n(q; z)| + |Xp,n(q; z)|.

Sinceqpr(1 + r)[n + p + 2]q

≤ r(1 + r)[n + p + 2]q

andqp+1[n]qz + [p + 1]q

[n + p + 2]q≤ r,

it follows

|Ep,n(q; z)| ≤ r(1 + r)[n + p + 2]q

|Dq(Dn,q(ep−1; z) − ep−1(z))| + r|Ep−1,n(q; z)| + |Xp,n(q; z)|.

Now we shall find the estimation of |E′p−1,n(q; z)| for p ≥ 1. Taking into account the fact that

Dn,q(ep−1; z) − ep−1(z) is a polynomial of degree ≤ p − 1, we have

|Dq(Dn,q(ep−1; z) − ep−1(z))| ≤ |(Dn,q(ep−1; z) − ep−1(z))′|≤p − 1

r||Dn,q(ep−1) − ep−1||r ≤ p − 1

r

(1 + r)[n + 1]q

p(p + 1)rp−2

≤ 2[n + 1]q

(p − 1)p(p + 1)rp−2.

Thusr(1 + r)

[n + p + 2]q|Dq(Dn,q(ep−1; z) − ep−1(z))| ≤ 4(p − 1)p(p + 1)rp

[n]2qand

|Ep,n(q; z)| ≤ 4(p − 1)p(p + 1)rp

[n]2q+ r|Ep−1,n(q; z)| + |Xp,n(q; z)|,

where

|Xp,n(q; z)| ≤rp−2

[n]2q(p(p − 1)(p − 2) + 5p3r + 4r2p2(p + 1)) + rp(1 − q)

≤ rp

[n]2qDp + rp(1 − q)

for all |z| ≤ r, p ≥ 1, n ∈ N, where

Dp = p(p − 1)(p − 2) + 5p3 + 4p2(p + 1).

Thus for all |z| ≤ r, p ≥ 1, n ∈ N,

|Ep,n(q; z)| ≤ r|Ep−1,n(q; z)| + rp

[n]2qFp,r + rp(1 − q),

where Fp,r is a polynomial of degree 3 in p defined as

Fp = Dp + 4(p − 1)p(p + 1).

74 N.I. MAHMUDOV, V. GUPTA

But E0,n(q; z) = 0, for any z ∈ C and therefore by writing last inequality for p = 1, 2, · · ·, weeasily obtain step by step the following

|Ep,n(q; z)| ≤ rp

[n]2q

p∑

j=1

Fj + rpp(1 − q) ≤ prp

[n]2qFp + rpp(1 − q).

We conclude that

∣∣Dn,q(f ; z) − f(z) − z(1 − z)f ′′(z) − zf ′(z)[n]q

∣∣∣

≤∞∑

p=1

|ap||Ep,n(q; z)| ≤ 1[n]2q

∞∑

p=1

|ap|pFprp + (1 − q)

∞∑

p=1

|ap|prp.

As f (4)(z) =∞∑

p=4app(p−1)(p−2)(p−3)zp−4 and the series is absolutely convergent in |z| ≤ r, it

easily follows that∞∑

p=4|ap|p(p−1)(p−2)(p−3)rp−4 < ∞, which implies that

∞∑p=1

|ap|pFprp < ∞.

This completes the proof of theorem. �

Remark. Let 0 < q < 1 be fixed. Since for n → ∞, we have 1[n]q

→ 1 − q, by passing tolimit with n → ∞ in the estimates in Theorem 5 we don’t obtain convergence of the operatorsDn,q(f ; z). But this situation can be improved by choosing 1 − 1

n2 ≤ qn < 1 with qn ↗ 1 asn → ∞. Indeed, since in this case 1

[n]qn→ 0 as n → ∞ and 1 − qn ≤ 1

n2 ≤ 1[n]2qn

from Theorem5, we get

∣∣∣Dn,qn(f ; z) − f(z) − z(1 − z)f ′′(z) + (1 − 2z)f ′(z)[n]qn

∣∣∣ ≤ Mr(f)[n]2qn

+1

[n]2qn

∞∑

p=1

|ap|prp,

that is the order of approximation 1[n]2qn

.

Acknowledgements. The authors are thankful to the referee for valuable suggestions leadingto overall improvements in the paper.

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