Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 454579, 14 pages
doi:10.1155/2012/454579
Research Article
Approximation Theorems for Generalized Complex
Kantorovich-Type Operators
N. I. Mahmudov and M. Kara
Eastern Mediterranean University, Gazimagusa, T.R. North Cyprus, Mersiin 10, Turkey
Correspondence should be addressed to N. I. Mahmudov, nazim.mahmudov@emu.edu.tr
Received 28 April 2012; Accepted 3 September 2012
Academic Editor: Jinyun Yuan
Copyright q 2012 N. I. Mahmudov and M. Kara. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The order of simultaneous approximation and Voronovskaja-type results with quantitative
estimate for complex q-Kantorovich polynomials q > 0 attached to analytic functions on compact
disks are obtained. In particular, it is proved that for functions analytic in {z ∈ C : |z| < R}, R > q,
the rate of approximation by the q-Kantorovich operators q > 1 is of order q−n versus 1/n for the
classical Kantorovich operators.
1. Introduction
For each integer k ≥ 0, the q-integer kq and the q-factorial kq ! are defined by
⎧
1 − qk
⎪
⎪
⎪
⎨ 1−q ,
kq :
⎪
⎪
⎪
⎩
k,
if q ∈ R \ {1},
for k ∈ N, 0q 0,
if q 1,
kq ! : 1q 2q · · · kq
1.1
for k ∈ N, 0! 1.
For integers 0 ≤ k ≤ n, the q-binomial coefficient is defined by
nq !
n
.
:
k q
kq !n − kq !
1.2
2
Journal of Applied Mathematics
For fixed 1 /
q > 0, we denote the q-derivative Dq fz of f by
Dq fz ⎧ f qz − fz
⎪
⎪
⎪
⎪
⎨ q − 1z ,
⎪
⎪
⎪
⎪
⎩f 0,
z/
0,
1.3
z 0.
Let DR be a disc DR : {z ∈ C : |z| < R} in the complex plane C. Denote by HDR the
m
space of all analytic functions on DR . For f ∈ HDR we assume that fz ∞
m0 am z .
In several recent papers, convergence properties of complex q-Bernstein polynomials,
proposed by Phillips 1, defined by
Bn,q
n
f
f; z k0
n−k−1
n
kq
n
k
j
pn,k q; z
1−q x x
f
k q
nq
nq
j0
k0
kq
1.4
and attached to an analytic function f in closed disks, were intensively studied by many
authors; see 2 and references their in. It is known that the cases 0 < q < 1 and q > 1 are
not similar to each other. This difference is caused by the fact that, for 0 < q < 1, Bn,q are
positive linear operators on C0, 1 while for q > 1, the positivity fails. The lack of positivity
makes the investigation of convergence in the case q > 1 essentially more difficult than
that for 0 < q < 1. There are few papers 3–6 studying systematically the convergence of
the q-Berntsein polynomials in the case q > 1. If q ≥ 1 then qualitative Voronovskaja-type
and saturation results for complex q-Bernstein polynomials were obtained in Wang and
Wu 5. Wu 6 studied saturation of convergence on the interval 0, 1 for the q-Bernstein
polynomials of a continuous function f for arbitrary fixed q > 1. On the other hand, Gal
7, 8, Anastassiou and Gal 9, 10, Mahmudov 11–13, and Mahmudov and Gupta 14
obtained quantitative estimates of the convergence and of the Voronovskaja’s theorem in
compact disks, for different complex Bernstein-Durrmeyer type operators.
The goal of the present note is to extend these type of results to complex Kantorovich
operators based on the q-integers, in the case q > 0, defined as follows:
n
pn,k q; z
Kn,q f; z k0
1 qk t q
dt.
f
1q
n
0
1.5
Notice that in the case q 1, these operators coincide with the classical Kantorovich
operators. For 0 < q ≤ 1 the operator Kn,q : C0, 1 → C0, 1 is positive and for q > 1,
it is not positive. The problems studied in this paper in the case q 1 were investigated in
2, 9.
We start with the following quantitative estimates of the convergence for complex qKantorovich-type operators attached to an analytic function in a disk of radius R > 1 and
center 0.
Journal of Applied Mathematics
3
Theorem 1.1. Let f ∈ HDR .
i Let 0 < q ≤ 1 and 1 ≤ r < R. For all z ∈ Dr and n ∈ N, one has
∞
−1 Kn,q f; z − fz ≤ 3 q
|am |mm 1r m .
2nq m1
1.6
ii Let 1 < q < R < ∞ and 1 ≤ r < R/q. For all z ∈ Dr and n ∈ N, one has
∞
2 |am |mm 1qm r m .
nq m1
Kn,q f; z − fz ≤
1.7
Remark 1.2. i Since nq → 1 − q−1 as n → ∞ in the estimate in Theorem 1.1i we do
not obtain convergence of Kn,q f; z to fz. But this situation can be improved by choosing
0 < q qn < 1 with qn 1 as n → ∞. Since in this case nqn → ∞ as n → ∞, from
Theorem 1.1i we get uniform convergence in Dr .
ii Theorem 1.1ii says that for functions analytic in DR , R > q, the rate of
approximation by the q-Kantorovich operators q > 1 is of order q−n versus 1/n for the
classical Kantorovich operators.
Let f ∈ HDR . Let us define
⎧
1 − z Dq fz − f z
⎪
1 − 2z ⎪
⎪
f
,
z
⎪
1 − q−1
⎨ 2
Lq f; z :
⎪
⎪
⎪
1 − 2z z1 − z ⎪
⎩
f z f z,
2
2
if |z| <
R
, R > q > 1,
q
1.8
if |z| < R, 0 < q ≤ 1.
It is not difficult to show that
∞
∞
mq − m m−1 1 − 2z Lq f; z q1 − z
z
am
am mzm−1
q
−
1
2
m1
m1
q
∞
am 1q · · · m − 1q z
m1
m−1
∞
1 − 2z am mzm−1 ,
1 − z 2 m1
1.9
q > 1.
Here we used the identity
mq − m
q−1
1q · · · m − 1q .
1.10
The next theorem gives Voronovskaja-type result in compact disks, for complex qKantorovich operators attached to an analytic function in DR , R > 1 and center 0.
4
Journal of Applied Mathematics
Theorem 1.3. Let f ∈ HDR .
i Let 0 < q ≤ 1 and 1 ≤ r < R. For all z ∈ Dr and n ∈ N one has
∞
1 − 2z z1 − z 28 q−1 f z −
f z ≤
|am |m2 m − 12 r m .
Kn,q f; z − fz −
2
2n 1q
2n 1q
2nq m2
1.11
ii Let 1 < q < R < ∞ and 1 ≤ r < R/q2 . For all z ∈ Dr and n ∈ N, one has
∞
14 1
Lq f; z ≤
|a |m2 m − 12 q2m r m .
Kn,q f; z − fz −
n2 m2 m
n 1q
q
1.12
Remark 1.4. i In the hypothesis on f in Theorem 1.3i choosing 0 < qn < 1 with qn 1 as
n → ∞, it follows that
1 − 2z z1 − z f z f z
lim n 1qn Kn,qn f; z − fz n→∞
2
2
1.13
uniformly in any compact disk included in the open disk DR .
ii Theorem 1.3ii gives explicit formulas of Voronovskaja-type for the q-Kantorovich
polynomials for q > 1.
iii Obviously the best order of approximation that can be obtained from the estimate
Theorem 1.3i is O1/n2qn and O1/n2 for q 1, while the order given by Theorem 1.3ii
is O1/q2n , q > 1, which is essentially better.
Next theorem shows that Lq f; z, q ≥ 1, is continuous about the parameter q for
f ∈ HDR , R > 1.
Theorem 1.5. Let R > 1 and f ∈ HDR . Then for any r, 0 < r < R,
lim Lq f; z L1 f; z
1.14
q → 1
uniformly on Dr .
As an application of Theorem 1.3, we present the order of approximation for complex
q-Kantorovich operators.
Theorem 1.6. Let 1 < q < R, 1 ≤ r < R/q2 (or 0 < q ≤ 1, 1 ≤ r < R) and f ∈ HDR . If f is not a
constant function then the estimate
Kn,q f − f ≥
r
1
Cr,q f ,
n 1q
n∈N
holds, where the constant Cr,q f depends on f, q and r but is independent of n.
1.15
Journal of Applied Mathematics
5
2. Auxiliary Results
Lemma 2.1. Let q > 0. For all n ∈ N, m ∈ N ∪ {0}, z ∈ C one has
Kn,q em ; z m m
j0
j
j
qj nq
Bn,q ej ; z ,
m
n 1q m − j 1
2.1
where em z zm .
Proof. The recurrence formula can be derived by direct computation.
Kn,q em ; z n
pn,k z
j0
k0
m m
j0
j
m m
j0
m 1 qj kj tm−j
q
m
j
n 1m
q
j
0
dt j
n
pn,k z
k0
m m
j0
j
j
qj kq
n 1m
q m−j 1
j
n k
qj nq
q
pn,k z
m
n 1q m − j 1 k0 njq
j
qj nq
Bn,q ej ; z .
m
n 1q m − j 1
2.2
Lemma 2.2. For all z ∈ Dr , r ≥ 1 one has
Kn,q em ; z ≤ r m ,
n, m ∈ N.
2.3
Proof. Indeed, using the inequality |Bn,q ej ; z| ≤ r j see 3, we get
j
m qj nq
m
Kn,q em ; z ≤
Bn,q ej ; z m
j
n 1q m − j 1
j0
m
m 1 qnq
1
m j j m
rm rm.
≤
q nq r j
n 1q
n 1m
q j0
2.4
Lemma 2.3. For all n, m ∈ N, z ∈ C, 1 /
q > 0 one has
Kn,q em1 ; z z1 − z
Dq Kn,q em ; z zKn,q em ; z
nq
1
n 1m1
q
m1
j0
jn 1q
1
m1 j j
Bn,q ej ; z .
q nq 1−
j
qm 1nq
m−j 2
2.5
6
Journal of Applied Mathematics
Proof. We know that see 2
z1 − z
Dq Bn,q ej ; z Bn,q ej1 ; z − zBn,q ej ; z .
nq
2.6
Taking the derivative of the formula 2.1 and using the above formula we have
j
m qj nq
z1 − z
z1 − z
m
Dq Kn,q em ; z Dq Bn,q ej ; z
m
j
nq
n
1
m
−
j
1
n
q
q
j0
m m
j
j0
m1
j1
j
qj nq
Bn,q ej1 ; z − zBn,q ej ; z
m
n 1q m − j 1
2.7
j−1
qj−1 nq
m
Bn,q ej ; z − zKn,q em ; z.
m
j − 1 n 1q m − j 2
It follows that
Kn,q em1 ; z z1 − z
Dq Kn,q em ; z zKn,q em ; z
nq
m1
j0
−
j−1
qj−1 nq
m
Bn,q ej ; z
m
j − 1 n 1q m − j 2
z1 − z
1
Dq Kn,q em ; z zKn,q em ; z m1
nq
n 1q m 2
2.8
m1
j1
q
m1
j1
j
qj nq
m1
Bn,q ej ; z
m1 j
m−j 2
n 1
j−1
m 1qnq − jn 1q
qj−1 nq
m1
Bn,q ej ; z
m
j
m 1n 1q
n 1q m − j 2
z1 − z
Dq Kn,q em ; z zKn,q em ; z
nq
m1
j0
j−1
m 1qnq − jn 1q
qj−1 nq
m1
Bn,q ej ; z .
m
j
m 1n 1q
n 1q m − j 2
Here we used the identity
m
j −1
j
m1
.
j
m 1
2.9
Journal of Applied Mathematics
7
For m ∈ N ∪ {0} define
En,m z :
⎧
z1 − z
1 − 2z
⎪
⎪
mzm−1 −
mm − 1zm−2 ,
Kn,q em ; z − em z −
⎪
⎪
2n
1
2n
1q
⎪
q
⎨
if 0 < q ≤ 1,
⎪
⎪
m−1
⎪
qzm−1 1 − z
1 − 2z
⎪
⎪
j q
mzm−1 −
,
⎩Kn,q em ; z − em z −
2n 1q
n 1q
j1
if q > 1.
2.10
Here it is assumed that
0
j1
jq 0.
Lemma 2.4. Let n, m ∈ N.
a If 0 < q < 1, one has the following recurrence formula:
En,m z z1 − z Dq Kn,q em−1 ; z − em−1 z zEn,m−1 z
nq
m − 1q
m−1
−
n 1q
nq
zm−1 1 − z −
1 − 2z m−1
z
2n 1q
2.11
j
m qj nq mqnq − jn 1q
1
m
Bn,q ej ; z .
m
j
mqn
n 1q j0
m−j 1
q
b If q > 1, one has
En,m z m − 1q m−1
z1 − z Dq Kn,q em−1 ; z − em−1 z zEn,m−1 z z 1 − z
nq
nq n 1q
−
j
m qj nq mqnq − jn 1q
1
1 − 2z m−1
m
z
Bn,q ej ; z .
m
2n 1q
mqnq
n 1q j0 j
m−j 1
2.12
Proof. We give the proof for the case q > 1. The case 0 < q < 1 is similar to that of q > 1.
b It is immediate that En,m z is a polynomial of degree less than or equal to m and
that En,0 z En,1 z 0.
8
Journal of Applied Mathematics
Using the formula 2.5, we get
En,m z m − 1q m−1
z1 − z Dq Kn,q em−1 ; z − em−1 z z 1 − z
nq
nq
⎛
⎞
m−2
qzm−2 1 − z
1
−
2z
⎠
z⎝En,m−1 z j q
m − 1zm−2 2n 1q
n 1q
j1
m−1
qzm−1 1 − z
1 − 2z
−
mzm−1 −
j q
2n 1q
n 1q
j1
2.13
j
m qj nq mqnq − jn 1q
1
m
Bn,q ej ; z .
m
mqnq
n 1q j0 j
m−j 1
A simple calculation leads us to the following relationship:
En,m z m − 1q m−1
z1 − z Dq Kn,q em−1 ; z − em−1 z zEn,m−1 z z 1 − z
nq
nq n 1q
−
j
m qj nq mqnq − jn 1q
1
1 − 2z m−1
m
z
Bn,q ej ; z ,
j
2n 1q
mqn
n 1m
m
−
j
1
q
q j0
2.14
which is the desired recurrence formula.
Remark 2.5. Lemmas 2.3 and 2.4 are true in the case q 1. In the formulae, we have to replace
q-derivative by the ordinary derivative.
3. Proofs of the Main Results
We give proofs for the case q > 1. The case 0 < q < 1 and q 1 are similar to that of q > 1.
Proof of Theorem 1.1. The use of the above recurrence we obtain the following relationship:
Kn,q em ; z − em z z1 − z
Dq Kn,q em−1 ; z z Kn,q em−1 ; z − em−1 z
nq
j
m qj nq mqnq − jn 1q
1
m
Bn,q ej ; z .
j
mqn
n 1m
m
−
j
1
q
q j0
3.1
Journal of Applied Mathematics
9
We can easily estimate the sum in the above formula as follows:
m j
j
1
1
m
j
j
1
−
B
−
;
z
e
q
n
n,q j
q
n 1m
j
m mqnq
m−j 1
q j0
⎛
⎞
j m−1
m−1
m−1
m − 1 m
qj nq q nq
j
j
1
⎠
m
⎝
≤
B
1
−
−
;
z
e
n,q j
m r
j
m
−
j
m
−
j
1
m
mqn
1
n 1m
n
q
q
q
j0
≤
m−1
qm−1 nm−1
2m qnq 1
q
n 1m
q
rm ≤
2m 1 m
r .
n 1q
3.2
It is known that by a linear transformation, the Bernstein inequality in the closed unit disk
becomes
Pm z ≤ m Pm ,
qr
qr
∀|z| ≤ qr, r ≥ 1,
3.3
where Pm qr max{|Pm z| : |z| ≤ qr} which combined with the mean value theorem in
complex analysis implies
P qz − Pm z m
Dq Pm ; z m
Pm qr ,
≤ Pm qr ≤
qz − z
qr
3.4
for all |z| ≤ r, where Pm z is a complex polynomial of degree ≤ m. From the above recurrence
formula 3.1, we get
Kn,q em ; z − em z ≤ |z||1 − z| Dq Kn,q em−1 ; z |z|Kn,q em−1 ; z − em−1 z 2m 1 r m
nq
n 1q
≤
r1 r m − 1 Kn,q em−1 r Kn,q em−1 ; z − em−1 z 2m 1 r m
qr
qr
nq
n 1q
2m − 1 m−1 m 2m 1 m
≤ r Kn,q em−1 ; z − em−1 z q r r
nq
n 1q
4m m m
≤ r Kn,q em−1 ; z − em−1 z q r .
nq
3.5
10
Journal of Applied Mathematics
By writing the last inequality for m 1, 2, . . ., we easily obtain, step by step, the following:
|Kn em ; z − em z| ≤
4m − 2 m−2 m−2
4
4m m m 4m − 1 m−1 m−1
q r r
q r
r2
q r
· · · r m−1
qr
nq
nq
nq
nq
4 m m
2mm 1 m m
q r m m − 1 · · · 1 ≤
q r .
nq
nq
3.6
Since Kn,q f; z is analytic in DR , we can write
∞
am Kn,q em ; z,
Kn,q f; z z ∈ DR ,
3.7
m0
which together with 3.6 immediately implies for all |z| ≤ r
∞
Kn,q f; z − fz ≤
|am |Kn,q em ; z − em z ≤
m0
∞
m
2 |cm |mm 1 qr .
nq m1
3.8
Proof of Theorem 1.3. A simple calculation and the use of the recurrence formula 2.5 lead us
to the following relationship:
En,m z m − 1q m−1
z1 − z Dq Kn,q em−1 ; z − em−1 z zEn,m−1 z z 1 − z
nq
nq n 1q
⎛
⎞
qm−1 nm−1
m
1
1
q
⎝1 −
⎠Bn,q em ; z
z − Bn,q em ; z n 1q
n 1q
n 1m−1
q
⎛
⎞
qm−1 nm−1
1
1
q
m−1
⎝
⎠Bn,q em−1 ; z B
−
1
;
z
−
z
e
n,q
m−1
2n 1q
2n 1q
n 1m−1
q
−
:
m − 1qm−2 nm−2
q
2n 1m
q
Bn,q em−1 ; z
m−2
m qj njq mqnq − jn 1q
1
Bn,q ej ; z
j
mqn
n 1m
m
−
j
1
q
q j0
9
k1
Ik .
3.9
Journal of Applied Mathematics
11
Firstly, we estimate I3 , I8 . It is clear that
|I3 | ≤
|I8 | ≤
m − 1q
nq n 1q
r m−1 1 r,
3.10
m − 1 m−1
m − 1 Bn,q em−1 ; z ≤
r .
2
2n 1q
2n 12q
Secondly, using the known inequality
1−
m
m
xk ≤
1 − xk ,
0 ≤ xk ≤ 1,
3.11
k1
k1
to estimate I5 , I6 , I9 .
⎛
⎞
qm−1 nm−1
1
q
⎝1 −
⎠Bn,q em ; z ≤ m − 1 r m ,
|I5 | ≤
m−1
n 1q
n 12q
n 1q
⎛
⎞
qm−1 nm−1
1
q
⎝1 −
⎠Bn,q em−1 ; z ≤ m − 1 r m−1 ,
|I6 | ≤
m−1
2n 1q
2n 12q
n 1q
|I9 | ≤
≤
j
m−2
m − 2
qj nq
1
mm − 1
j
n 1m
m−j m−j −1 m−j 1
q j0
2mm − 1n 1m−2
q
n 1m
q
rm 2mm − 1
n 12q
1−
j
j
−
m mqnq
3.12
rj
rm.
Finally, we estimate I4 , I7 . We use 2, Theorem 1.1.2
|I4 | |I7 | ≤
≤
m
1
1
z − Bn,q em ; z Bn em−1 ; z − zm−1 2n 1q
n 1q
2m − 1q m − 1
nq n 1q
rm m − 2q m − 2
nq n 1q
3.13
r m−1 .
12
Journal of Applied Mathematics
Using 3.6, 3.10, 3.12, and 3.13 in 3.9 finally we have m ≥ 3
|En,m z| ≤
m − 1q m−1
r1 r Dq Kn,q em−1 ; z − em−1 z r|En,m−1 z| r 1 r
nq
nq n 1q
2m − 1q m − 1
nq n 1q
m − 1
2n 12q
r m−1 rm m−1
n 2mm − 1
n 12q
12q
rm m−1
2n 12q
r m−1 m − 2q m − 2
nq n 1q
rm
≤
10mm − 1q m
r1 r m − 1 Kn,q em−1 − em−1 r|En,m−1 z| r
qr
qr
nq
n2q
≤
10mm − 1q m
m − 11 r 2m − 1m 2m−1 m−1
q
r
r|En,m−1 z| r
nq
nq
n2q
≤ r|En,m−1 z| ≤ r|En,m−1 z| 4mm − 12
n2q
q2m r m 14m2 m − 12
n2q
r m−1
10mm − 1
n2q
qm r m
q2m r m .
3.14
As a consequence, we get
|En,m z| ≤
14m2 m − 12
n2q
q2m r m .
3.15
This inequality combined with
∞
1
Lq f, z ≤
|a ||E z|
Kn,q f; z − fz −
m1 m n,m
n 1q
3.16
immediately implies the required estimate in statement.
m−4
Note that since f 4 ∞
and the series is absolutely
m4 am mm − 1m − 2m − 3z
convergent for all |z| < R, it easily follows the finiteness of the involved constants in the
statement.
Proof of Theorem 1.6. For all z ∈ DR and n ∈ N, we get
Kn,q
f; z − fz 1
1
.
Lq f; z n 1q Kn,q f; z − fz −
Lq f; z
n 1q
n 1q
3.17
Journal of Applied Mathematics
13
We apply
F G
r ≥ |
F
r − G
r | ≥ F
r − G
r
3.18
to get
Kn,q f − f ≥
r
1
1
Lq f; z − n 1 K f; z − fz −
Lq f; z .
q
r
n,q
n 1q
n 1q
r
3.19
Because by hypothesis f is not a constant in DR , it follows Lq f; z
r > 0. Indeed, assuming
the contrary, it follows that Lq f; z 0 for all z ∈ DR that is
∞
am
m1
∞
m−1
1
− z mzm−1 am
j q zm−1 1 − z 0,
2
m1
j1
⎞
m
m−1
1
1
⎝ m 1am1 − am am1
a1 a1 j q − am
j q ⎠zm 0
2
2
m1
j1
j1
∞
⎛
3.20
for all z ∈ DR \ {0}. Thus am 0, m 1, 2, 3, . . .. Thus, f is constant, which is contradiction
with the hypothesis.
Now, by Theorem 1.3, we have
1
Lq f; z n 1q Kn,q f; z − fz −
n 1q
3.21
∞
n 1q 14 ≤
|am |m2 m − 12 q2m r m −→ 0
nq nq m2
as n −→ ∞.
Consequently, there exists n1 depending only on f and r such that for all n ≥ n1 we have
1 1
Lq f; z − n 1 K f; z − fz −
Lq f; z ≥ Lq f; z r ,
q
r
n,q
2
n 1q
3.22
r
which implies
Kn,q f − f ≥
r
1
1
Lq f; z ,
r
n 1q 2
∀n ≥ n1 .
3.23
For 1 ≤ n ≤ n1 − 1, we have
Kn,q f − f ≥
r
1
1
Mr,n f > 0,
n 1q Kn,q f − f r n 1q
n 1q
3.24
14
Journal of Applied Mathematics
which finally implies that
Kn,q f − f ≥
r
1
Cr,q f ,
n 1q
3.25
for all n, with Cr,q f min{Mr,1 f, . . . , Mr,n1 −1 f, 1/2
Lq f; z
r }.
Proof of Theorem 1.5. Proof is similar to that of Theorem 1.3 5.
References
1 G. M. Phillips, “Bernstein polynomials based on the q-integers,” Annals of Numerical Mathematics, vol.
4, no. 1–4, pp. 511–518, 1997.
2 S. G. Gal, Approximation by complex Bernstein and convolution type operators, vol. 8, World Scientific,
2009.
3 S. Ostrovska, “q-Bernstein polynomials and their iterates,” Journal of Approximation Theory, vol. 123,
no. 2, pp. 232–255, 2003.
4 S. Ostrovska, “The sharpness of convergence results for q-Bernstein polynomials in the case q > 1,”
Czechoslovak Mathematical Journal, vol. 58, no. 4, pp. 1195–1206, 2008.
5 H. Wang and X. Wu, “Saturation of convergence for q-Bernstein polynomials in the case q > 1,” Journal
of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 744–750, 2008.
6 Z. Wu, “The saturation of convergence on the interval 0, 1 for the q-Bernstein polynomials in the case
q > 1,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 137–141, 2009.
7 S. G. Gal, “Voronovskaja’s theorem and iterations for complex Bernstein polynomials in compact
disks,” Mediterranean Journal of Mathematics, vol. 5, no. 3, pp. 253–272, 2008.
8 S. G. Gal, “Approximation by complex genuine Durrmeyer type polynomials in compact disks,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 1913–1920, 2010.
9 G. A. Anastassiou and S. G. Gal, “Approximation by complex Bernstein-Schurer and KantorovichSchurer polynomials in compact disks,” Computers & Mathematics with Applications, vol. 58, no. 4, pp.
734–743, 2009.
10 G. A. Anastassiou and S. G. Gal, “Approximation by complex bernstein-durrmeyer polynomials in
compact disks,” Mediterranean Journal of Mathematics, vol. 7, no. 4, pp. 471–482, 2010.
11 N. I. Mahmudov, “Approximation by genuine q-Bernstein-Durrmeyer polynomials in compact
disks,” Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 1, pp. 77–89, 2011.
12 N. I. Mahmudov, “Convergence properties and iterations for q-Stancu polynomials in compact disks,”
Computers & Mathematics with Applications. An International Journal, vol. 59, no. 12, pp. 3763–3769, 2010.
13 N. I. Mahmudov, “Approximation by Bernstein-Durrmeyer-type operators in compact disks,” Applied
Mathematics Letters, vol. 24, no. 7, pp. 1231–1238, 2011.
14 N. I. Mahmudov and V. Gupta, “Approximation by genuine Durrmeyer-Stancu polynomials in compact disks,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 278–285, 2012.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014