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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: [email protected] of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka New Delhi-110078, India
(E-mail: [email protected]
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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