Approximation by Complex Bernstein-Durrmeyer Polynomials in Compact Disks

Mediterr. J. Math. 7 (2010), 471–482DOI 10.1007/s00009-010-0036-11660-5446/10/040471-12, published online March 23, 2010© 2010 Birkhäuser / Springer Basel AG

Mediterranean Journalof Mathematics

Approximation by ComplexBernstein-Durrmeyer Polynomials inCompact Disks

George A. Anastassiou∗ and S.G. Gal

Abstract. In this paper, the order of simultaneous approximation andVoronovskaja kind results with quantitative estimate for complex Bern-stein-Durrmeyer polynomials attached to analytic functions on compactdisks are obtained. In this way, we put in evidence the overconvergencephenomenon for Bernstein-Durrmeyer polynomials, namely the exten-sions of approximation properties (with quantitative estimates) fromreal intervals to compact disks in the complex plane.

Mathematics Subject Classification (2010). Primary 30E10 ; Secondary41A25.

Keywords. Complex Bernstein-Durrmeyer polynomials, simultaneous a-pproximation, Voronovskaja-type result.

1. Introduction

Concerning the convergence of Bernstein polynomials in the complex plane,Bernstein proved (see e.g. [12, p. 88]) that if f : G → C is analytic in the openset G ⊂ C, with D1 ⊂ G (where D1 = {z ∈ C : |z| < 1}), then the complexBernstein polynomials Bn(f)(z) =

∑nk=0

(nk

)zk(1 − z)n−kf(k/n), uniformly

converge to f in D1.Exact estimates of order O(1/n) of this uniform convergence and, in

addition, of the simultaneous approximation, were proved in [6] and [7]. In [8]a Voronovskaja-type result with quantitative estimate for complex Bernsteinpolynomials in compact disks was obtained.

This paper was written during the 2009 Spring Semester when the second author was a Vis-iting Professor at the Department of Mathematical Sciences, The University of Memphis,TN, U.S.A.∗Corresponding author.

472 G.A. Anastassiou and S.G. Gal Mediterr. J. Math.

In [9]-[11] similar results for Bernstein-Stancu and Kantorovich-Stancupolynomials were obtained while in [2] similar results for Bernstein-Schurerpolynomials were proved.

The goal of this paper is to obtain approximation results for the complexBernstein-Durrmeyer polynomials (introduced and studied in the case of realvariable in [3-5]) defined by

Dn(f)(z) = (n + 1)n∑

k=0

(n

k

)zk(1 − z)n−k

∫ 1

0

(n

k

)tk(1 − t)n−kf(t)dt, z ∈ C.

Note that all these results put in evidence the overconvergence phenomenonfor Bernstein-Durrmeyer polynomials, that is the extensions of approximationproperties (with quantitative estimates) from real intervals to compact disksin the complex plane.

2. Approximation by Complex Bernstein-DurrmeyerPolynomials

For our reasonings will be very useful the following well-known formula (seee.g. [1, p. 133])

Dn(ep)(z) =(n + 1)!

(n + 1 + p)!

min{n,p}∑k=0

(p

k

)p!k!

· n!(n − k)!

zk

=1(

n+1+pn+1

) ·min{n,p}∑

k=0

(p

k

)(n

k

)zk, (2.1)

for all p, n ∈ N⋃{0} and z ∈ C, where ep = zp.

Also, the following lemma holds.

Lemma 2.1. For all p, n ∈ N⋃{0} the inequality

1(n+1+p

n+1

) ·min{n,p}∑

k=0

(p

k

)(n

k

)=

n + 1n + 1 + p

≤ 1,

holds.

Proof. From the formula of definition we have for all n, p

Dn(ep)(1) = (n + 1)∫ 1

0tn+pdt =

n + 1n + 1 + p

≤ 1,

which combined with (2.1) written for z = 1 immediately implies the inequal-ity in the statement. �

The first main result is expressed by the following upper estimates.

Corollary 2.2. Let r ≥ 1.(i) For all p, n ∈ N

⋃{0} and |z| ≤ r we have |Dn(ep)(z)| ≤ rp.

Vol. 7 (2010) Complex Bernstein-Durrmeyer Polynomials 473

(ii) Let f(z) =∑∞

k=0 ckzk for all |z| < R and take 1 ≤ r < R. For all|z| ≤ r and n ∈ N we have

|Dn(f)(z) − f(z)| ≤ Mr(f)n

,

where Mr(f) = 2∑∞

p=1 |cp|p(p + 1)rp.

Proof. (i) By (2.1) and Lemma 2.1 we obtain

|Dn(ep)(z)|

≤ 1(n+1+p

n+1

) ·min{n,p}∑

k=0

(p

k

)(n

k

)rk ≤ rp 1(

n+1+pn+1

) ·min{n,p}∑

k=0

(p

k

)(n

k

)≤ rp,

which proves (i).(ii) First we prove that Dn(f)(z) =

∑∞k=0 ckDn(ek)(z), for all |z| ≤ r.

Indeed, denoting fm(z) =∑m

j=0 cjzj, |z| ≤ r, m ∈ N, since from the lin-

earity of Dn we obviously have Dn(fm)(z) =∑m

k=0 ckDn(ek)(z), it suf-fices to prove that for any fixed n ∈ N and |z| ≤ r with r ≥ 1, we havelimm→∞Dn(fm)(z) = Dn(f)(z). But this is immediate from limm→∞ ‖fm −f‖r = 0 (where ‖f‖r = max{|f(z)|; |z| ≤ r} ) and from the inequality

|Dn(fm)(z) − Dn(f)(z)|

≤ (n + 1)n∑

k=0

(n

k

)|zk(1 − z)n−k|

∫ 1

0

(n

k

)|tk(1 − t)n−k| · |fm(t) − f(t)|dt

≤n∑

k=0

(n

k

)|zk(1 − z)n−k| · ‖fm − f‖r ≤ Mr,n‖fm − f‖r,

valid for all |z| ≤ r.Therefore we get

|Dn(f)(z) − f(z)| ≤∞∑

p=0

|cp| · |Dn(ep)(z) − ep(z)|

=∞∑

p=1

|cp| · |Dn(ep)(z) − ep(z)|,

since Dn(e0) = e0.We have two cases : 1) 1 ≤ p ≤ n ; 2) p > n.Case 1). From (2.1) and Lemma 2.1 we obtain

Dn(ep)(z) − ep(z) = zp

( (np

)(n+1+p

n+1

) − 1

)+

1(n+1+p

n+1

) p−1∑j=0

(p

j

)(n

j

)zj,


and

|Dn(ep)(z) − ep(z)| ≤ rp

[1 −

(np

)(n+1+p

n+1

)]

+ rp

[n + 1

n + 1 + p−

(np

)(n+1+p

n+1

)]

≤ 2rp

[1 −

(np

)(n+1+p

n+1

)]

.

Here it is easy to see that we can write(np

)(n+1+p

n+1

) =

(np

)(n+1+p

p

) = Πpj=1

n − p + j

n + j + 1.

But applying the inequality (easily proved by mathematical induction)

1 − Πkj=1xj ≤

k∑j=1

(1 − xj), 0 ≤ xj ≤ 1, j = 1, ..., k,

for xj = n−p+jn+j+1 and k = p, we obtain

1 − Πpj=1

n − p + j

n + j + 1≤

p∑j=1

(1 − n − p + j

n + j + 1

)= (p + 1)

p∑j=1

1n + j + 1

≤ p(p + 1)n + 2

.

Therefore it follows

|Dn(ep)(z) − ep(z)| ≤ 2p(p + 1)rp

n + 2≤ 2p(p + 1)rp

n.

Case 2). By (i) and by p > n we obtain

|Dn(ep)(z) − ep(z)| ≤ |Dn(ep)(z)| + |ep(z)| ≤ 2rp <2p

nrp.

In conclusion, from both cases (i) and (ii) we obtain for all p, n ∈ N

|Dn(ep)(z) − ep(z)| ≤ 2p(p + 1)rp

n,

which implies

|Dn(f)(z) − f(z)| ≤ 2n

∞∑p=1

|cp|p(p + 1)rp

and proves the theorem. �

In what follows we look for a Voronovskaja type result with a quan-titative estimate. For this purpose first we need a recurrence formula forDn(ep)(z).

Lemma 2.3. For all p, n ∈ N⋃{0} and z ∈ C we have

Dn(ep+1)(z) =z(1 − z)n + p + 2

D′n(ep)(z) +nz + p + 1n + p + 2

Dn(ep)(z).


Proof. By simple calculation we obtain

D′n(ep)(z) = (n + 1)n∑

k=0

(n

k

)kzk−1(1 − z)n−k

∫ 1

0

(n

k

)tk(1 − t)n−ktpdt

− (n + 1)n∑

k=0

(n

k

)(n − k)zk(1 − z)n−k−1

∫ 1

0

(n

k

)tk(1 − t)n−ktpdt

=1z

[(n + 1)

n∑k=0

(n

k

)zk(1 − z)n−k

∫ 1

0

(n

k

)ktk(1 − t)n−ktpdt

]

− n

1 − z· Dn(ep)(z) (2.2)

+1

1 − z(n + 1)

n∑k=0

(n

k

)zk(1 − z)n−k

∫ 1

0

(n

k

)ktk(1 − t)n−ktpdt

=1

z(1 − z)

[(n + 1)

n∑k=0

(n

k

)zk(1 − z)n−k

∫ 1

0

(n

k

)ktp+k(1 − t)n−kdt

]

− n

1 − zDn(ep)(z). (2.3)

Denoting the Euler’s Beta function B(p, q) =∫ 10 tp−1(1 − t)q−1dt it is

known that we have the formula (see e.g. [1, p. 132]) B(p, q) = (p−1)!(q−1)!(p+q−1)! .

This allows to immediately prove the relationship

(n+p+2)(

n

k

)B(p+k+2, n−k+1) = (p+1+k)

(n

k

)B(p+k+1, n−k+1),

which is equivalent to

(n+p+2)∫ 1

0

(n

k

)tp+1+k(1− t)n−kdt = (p+1+k)

∫ 1

0

(n

k

)tp+k(1− t)n−kdt,

and to∫ 1

0

(n

k

)ktp+k(1 − t)n−kdt = (n + p + 2)

∫ 1

0

(n

k

)tp+1+k(1 − t)n−kdt

− (p + 1)∫ 1

0

(n

k

)tp+k(1 − t)n−kdt.

Using this last formula in (2.2) we obtain

D′n(ep)(z)

=1

z(1 − z)(n + p + 2)Dn(ep+1)(z) − p + 1

z(1 − z)Dn(ep)(z) − n

1 − zDn(ep)(z),

which implies the recurrence in the statement. �

In the real case, the following Voronovskaja’s theorem is well-known

limn→∞

[Dn(f)(x) − f(x) − [x(1 − x)f ′(x)]′

n

]= 0


for all x ∈ [0, 1].In what follows we will prove the Voronovskaja theorem with a quanti-

tative estimate for the complex version of Bernstein-Durrmeyer polynomials,as follows.

Theorem 2.4. Let R > 1 and suppose that f : DR → C is analytic in DR ={z ∈ C; |z| < R}, that is we can write f(z) =

∑∞k=0 ckzk, for all z ∈ DR.

For any fixed r ∈ [1, R) and for all n ∈ N, |z| ≤ r, the following Voro-novskaja-type result holds∣∣∣∣Dn(f)(z) − f(z) − [z(1 − z)f ′(z)]′

n

∣∣∣∣ ≤ Mr(f)n2 ,

where Mr(f) =∑∞

k=1 |ck|kBk,rrk−1 < ∞ and

Bk,r = 2(k − 1)3 + r(4k3 + 6k2 + 6k + 2) + 2r2k(k2 + 1) + 4k(k − 1)2(1 + r).

Proof. Denoting ek(z) = zk, k = 0, 1, ..., and πk,n(z) = Dn(ek)(z), by theproof of Corollary 2.2, (ii) we can write Dn(f)(z) =

∑∞k=0 ckπk,n(z). Also,

since

[z(1 − z)f ′(z)]′

n=

(z(1 − z)

n

∞∑k=1

kckzk−1

)′=

( ∞∑k=1

kck

nzk(1 − z)

)′

=∞∑

k=1

kck

n[kzk−1(1 − z) − zk] =

∞∑k=1

ckkzk−1

n[k − (k + 1)z],

this immediately implies∣∣∣∣Dn(f)(z) − f(z) − [z(1 − z)f ′(z)]′

n

∣∣∣∣≤

∞∑k=1

|ck| ·∣∣∣∣πk,n(z) − ek(z) − kzk−1[k − (k + 1)z]

n

∣∣∣∣ ,for all z ∈ D1, n ∈ N.

In what follows, we will use the recurrence obtained in Lemma 2.3

πk+1,n(z) =z(1 − z)n + k + 2

π′k,n(z) +nz + k + 1n + k + 2

πk,n(z),

for all n ∈ N, z ∈ C and k = 0, 1, ....If we denote

Ek,n(z) = πk,n(z) − ek(z) − kzk−1[k − (k + 1)z]n

,

then it is clear that Ek,n(z) is a polynomial of degree ≤ k and by a sim-ple calculation and the use of the above recurrence we obtain the followingrelationship

Ek,n(z) =z(1 − z)n + k + 1

E′k−1,n(z) +nz + k

n + k + 1Ek−1,n(z) + Xk,n(z),


where

Xk,n(z) =zk−2

n(n + k + 1){2(k− 1)3 + z[−4k3 +6k2− 6k+2]+ z2[2k(k2 +1)]},

for all k ≥ 1, n ∈ N and |z| ≤ r.We will use the estimate obtained in the proof of Corollary 2.2,

|πk,n(z) − ek(z)| ≤ 2k(k + 1)rk

n,

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

|Ek,n(z)| ≤ r(1 + r)n + k + 1

|E′k−1,n(z)| + nr + k

n + k + 1|Ek−1,n(z)| + |Xk,n(z)|

Since r(1+r)n+k+1 ≤ r(1+r)

n and nr+kn+k+1 ≤ r it follows

|Ek,n(z)| ≤ r(1 + r)n

|E′k−1,n(z)| + r|Ek−1,n(z)| + |Xk,n(z)|.Now we will estimate |E′k−1,n(z)|, for k ≥ 1. Taking into account thatEk−1,n(z) is a polynomial of degree ≤ (k − 1), we obtain

|E′k−1,n(z)| ≤ k − 1r

‖Ek−1,n(z)‖r

≤ k − 1r

[‖πk−1,n − ek−1‖r +

∥∥∥∥ (k − 1)ek−2[k − 1 − ke1]n

∥∥∥∥r

]

≤ k − 1r

[2(k − 1)krk−1

n+

rk−2(r + 1)(k − 1)kn

]

≤ k(k − 1)2

n

[2rk−2 +

r + 1r

rk−2]≤ k(k − 1)2

n

[2rk−2 + 2rk−2]

=4k(k − 1)2rk−2

n.

This implies

r(1 + r)n

|E′k−1,n(z)| ≤ 4k(k − 1)2(1 + r)rk−1

n2 ,

and

|Ek,n(z)| ≤ r|Ek−1,n(z)| + 4k(k − 1)2(1 + r)rk−1

n2 + |Xk,n(z)|,where

|Xk,n(z)| ≤ rk−2

n2 {2(k − 1)3 + r(4k3 + 6k2 + 6k + 2) + 2r2k(k2 + 1)}

≤ rk−1

n2 Ak,r,

for all |z| ≤ r, k ≥ 1, n ∈ N, where

Ak,r = 2(k − 1)3 + r(4k3 + 6k2 + 6k + 2) + 2r2k(k2 + 1).


Denoting now

Bk,r = Ak,r + 4k(k − 1)2(1 + r),

we obtain

|Ek,n(z)| ≤ r|Ek−1,n(z)| + rk−1

n2 Bk,r,

for all |z| ≤ r, k ≥ 1, n ∈ N, where Bk,r is a polynomial of degree 3 in k.But E0,n(z) = 0, for any z ∈ C and therefore by writing the last in-

equality for k = 1, 2, ..., we easily obtain, step by step the following

|Ek,n(z)| ≤ rk−1

n2

⎡⎣ k∑

j=1

Bj,r

⎤⎦ ≤ krk−1

n2 Bk,r.

As a conclusion, we obtain∣∣∣∣Dn(f)(z) − f(z)− [z(1 − z)f ′(z)]′

n

∣∣∣∣ ≤∞∑

k=1

|ck| · |Ek,n(z)|

≤ 1n2

∞∑k=1

|ck|kBk,rrk−1.

Note that since f (4)(z) =∑∞

k=4 ckk(k − 1)(k − 2)(k − 3)zk−4, and theseries is absolutely convergent in |z| ≤ r, it easily follows that

∑∞k=4 |ck|k(k−

1)(k − 2)(k − 3)rk−4 < ∞, which implies that∑∞

k=1 |ck|kBk,rrk−1 < ∞ and

proves the theorem. �

Finally we will obtain the exact order in approximation by complexBernstein-Durrmeyer polynomials and their derivatives. In this sense wepresent the following results.

Theorem 2.5. Let R > 1, DR = {z ∈ C; |z| < R} and let us suppose thatf : DR → C is analytic in DR, that is we can write f(z) =

∑∞k=0 ckzk, for all

z ∈ DR. If f is not a polynomial of degree 0, then for any r ∈ [1, R) we have

‖Dn(f) − f‖r ≥ Cr(f)n

, n ∈ N,

where the constant Cr(f) depends only on f and r.

Proof. For all z ∈ DR and n ∈ N we have

Dn(f)(z) − f(z)

=1n

{[z(1 − z)f ′(z)]′ +

1n

[n2

(Dn(f)(z) − f(z) − [z(1 − z)f ′(z)]′

n

)]}.

In what follows we will apply to this identity the following obvious property:

‖F + G‖r ≥ | ‖F‖r − ‖G‖r | ≥ ‖F‖r − ‖G‖r.


It follows

‖Dn(f) − f‖r

≥ 1n

{‖[e1(1 − e1)f ′]′‖r −

1n

[n2

∥∥∥∥Dn(f) − f − [e1(1 − e1)f ′]′

n

∥∥∥∥r

]}.

Taking into account that by hypothesis f is not a polynomial of degree 0 inDR, we get ‖[e1(1 − e1)f ′]′‖r > 0. Indeed, supposing the contrary it followsthat [z(1 − z)f ′(z)]′ = 0 for all z ∈ Dr, which implies z(1 − z)f ′(z) = C forall |z| ≤ r, that is f ′(z) = C

z(1−z) for all |z| ≤ r. But since f is analytic inDr we necessarily have C = 0, which implies f ′(z) = 0 and f(z) = c for allz ∈ Dr, a contradiction. But by Theorem 2.4 we have

n2∥∥∥∥Dn(f) − f − [e1(1 − e1)f ′]′

n

∥∥∥∥r

≤ Mr(f).

Therefore, there exists an index n0 depending only on f and r, such that forall n ≥ n0 we have

‖[e1(1 − e1)f ′]′‖r −1n

[n2

∥∥∥∥Dn(f) − f − [e1(1 − e1)f ′]′

n

∥∥∥∥r

]

≥ 12‖[e1(1 − e1)f ′]′‖r ,

which immediately implies

‖Dn(f) − f‖r ≥ 1n· 12‖[e1(1 − e1)f ′]′‖r , ∀n ≥ n0.

For n ∈ {1, ..., n0 − 1} we obviously have ‖Dn(f) − f‖r ≥ Mr,n(f)n with

Mr,n(f) = n · ‖Dn(f)− f‖r > 0. Indeed, if we would have ‖Dn(f)− f‖r = 0would follow Dn(f)(z) = f(z) for all |z| ≤ r, which is valid only for f aconstant function, contradicting the hypothesis on f .

Therefore, finally we get ‖Dn(f)−f‖r ≥ Cr(f)n for all n, where Cr(f) =

min{Mr,1(f), ..., Mr,n0−1(f), 12 ‖[e1(1 − e1)f ′]′‖r}, which ends the proof. �

Combining now Corollary 2.5 with Corollary 2.2 (ii) we immediately getthe following.

Corollary 2.6. Let R > 1, DR = {z ∈ C; |z| < R} and let us suppose thatf : DR → C is analytic in DR. If f is not a polynomial of degree 0, then forany r ∈ [1, R) we have

‖Dn(f) − f‖r ∼ 1n

, n ∈ N,

where the constants in the equivalence depend only on f and r.

For the derivatives of complex Bernstein-Durrmeyer polynomials we canstate the following result.

Theorem 2.7. Let DR = {z ∈ C; |z| < R} be with R > 1 and let us supposethat f : DR → C is analytic in DR, i.e. f(z) =

∑∞k=0 ckzk, for all z ∈ DR.


Also, let 1 ≤ r < r1 < R and p ∈ N be fixed. If f is not a polynomial ofdegree ≤ p − 1, then we have

‖D(p)n (f) − f (p)‖r ∼ 1

n,

where the constants in the equivalence depend only on f , r, r1 and p.

Proof. Denoting by Γ the circle of radius r1 and center 0 (where r1 > r ≥ 1),by the Cauchy’s formulas it follows that for all |z| ≤ r and n ∈ N we have

D(p)n (f)(z) − f (p)(z) =

p!2πi

∫Γ

Dn(f)(v) − f(v)(v − z)p+1 dv,

which by Corollary 2.2 (ii) and by the inequality |v − z| ≥ r1 − r valid for all|z| ≤ r and v ∈ Γ, immediately implies

‖D(p)n (f) − f (p)‖r ≤ p!

2π· 2πr1

(r1 − r)p+1 ‖Dn(f) − f‖r1 ≤ Mr1(f)p!r1

n(r1 − r)p+1 .

It remains to prove the lower estimate for ‖D(p)n (f) − f (p)‖r.

For this purpose, as in the proof of Theorem 2.5, for all v ∈ Γ and n ∈ N

we have

Dn(f)(v) − f(v)

=1n

{[v(1 − v)f ′(v)]′ +

1n

[n2

(Dn(f)(v) − f(v) − [v(1 − v)f ′(v)]′

n

)]},

which replaced in the above Cauchy’s formula implies

D(p)n (f)(z) − f (p)(z) =

1n

{[z(1 − z)f ′(z)](p+1)

+1n· p!2πi

∫Γ

n2(Dn(f)(v) − f(v) − [v(1−v)f ′(v)]′

n

)(v − z)p+1 dv

}

=1n

{[z(1 − z)f ′(z)](p+1)

+1n· p!2πi

∫Γ

n2(Dn(f)(v) − f(v) − [v(1−v)f ′(v)]′

n

)(v − z)p+1 dv

}.

Passing now to ‖ · ‖r it follows

‖D(p)n (f) − f (p)‖r ≥ 1

n

{∥∥∥[e1(1 − e1)f ′](p+1)

∥∥∥r

− 1n

∥∥∥∥∥∥p!2π

∫Γ

n2(Dn(f)(v) − f(v) − [v(1−v)f ′(v)]′

n

)(v − z)p+1 dv

∥∥∥∥∥∥r

},


where by using Theorem 2.4 we get∥∥∥∥∥∥p!2π

∫Γ

n2(Dn(f)(v) − f(v) − [v(1−v)f ′(v)]′

n

)(v − z)p+1 dv

∥∥∥∥∥∥r

≤ p!2π

· 2πr1n2

(r1 − r)p+1

∥∥∥∥Dn(f) − f − [e1(1 − e1)f ′]′

n

∥∥∥∥r1

≤ Mr1(f)p!r1

(r1 − r)p+1 .

But by hypothesis on f we have∥∥∥[e1(1 − e1)f ′]

(p+1)∥∥∥

r> 0.

Indeed, supposing the contrary it follows that z(1 − z)f ′(z) is a poly-nomial of degree ≤ p. For p = 1 we get z(1 − z)f ′(z) = Az + B, whichimplies f ′(z) = Az+B

z(1−z) . Because f ′(z) is analytic function, it necessarily fol-lows A = B = 0, that is f ′(z) = 0, for all |z| ≤ r and therefore f is apolynomial of degree 0, contradicting the hypothesis. If p ≥ 2, then from theidentity z(1 − z)f ′(z) = Qp(z) for all |z| ≤ r (where Qp(z) is a polynomialof degree ≤ p in z), it easily follows that f ′(z) is a polynomial of degree≤ p − 2, that is f(z) is a polynomial of degree ≤ p − 1, contradicting againthe hypothesis.

In continuation, taking also into account that because f is not a poly-nomial of degree ≤ p− 1 we have ‖[Dn(f)− f ](p)‖r > 0, reasoning exactly asin the proof of Theorem 2.5, we easily get the desired conclusion. �

References

[1] O. Agratini, Approximation by Linear Operators (in Romanian), Cluj Univer-sity Press, Cluj-Napoca, 2000.

[2] G. A. Anastassiou and S. G. Gal, Approximation by complex Bernstein-Schurerand Kantorovich-Schurer polynomials in compact disks, Computers and Math-ematics with Applications, 58(2009), No. 4, 734-743.

[3] M. M. Derriennic, Sur l’approximation des fonctions d’une ou plusieurs vari-ables par des polynomes de Bernstein modifies et application au probleme demoments, These de 3e cycle, Univ. de Rennes, 1978.

[4] M. M. Derriennic, Sur l’approximation des fonctions integrable sur [0, 1] pardes polynomes de Bernstein modifies, J. Approx. Theory, 31(1981), 325-343.

[5] J. L. Durrmeyer, Une formule d’inversion de la transformee de Laplace. Ap-plications a la theorie des moments, These de 3e cycle, Fac. des Sciences del’Universite de Paris, 1967.

[6] S. G. Gal, Exact orders in simultaneous approximation by complex Bernsteinpolynomials, J. Concr. Applic. Math., 7(2009), No. 3, 215-220.

[7] S. G. Gal, Shape Preserving Approximation by Real and Complex Polynomials,Birkhauser Publ., Boston-Basel-Berlin, 2008.

[8] S. G. Gal, Voronovskaja’s theorem and iterations for complex Bernstein poly-nomials in compact disks, Mediterr. J. of Math., 5(2008), No. 3, 253-272.

[9] S. G. Gal, Approximation by complex Bernstein-Kantorovich and Stancu-Kantorovich polynomials and their iterates in compact disks, Rev. Anal. Numer.Theor. Approx. (Cluj), 37(2008), No. 2, 159-168.


[10] S. G. Gal, Exact orders in simultaneous approximation by complex Bernstein-Stancu polynomials, Revue Anal. Numer. Theor. Approx. (Cluj), 37(2008), No.1, 47-52.

[11] S. G. Gal, Approximation by complex Bernstein-Stancu polynomials in compactdisks, Results in Mathematics, 53(2009), No. 3-4, 245-256.

[12] G. G. Lorentz, Bernstein Polynomials, 2nd edition, Chelsea Publ., New York,1986.

Acknowledgment

The authors thank the referee for the useful remarks.

George A. AnastassiouDepartment of Mathematical SciencesThe University of MemphisMemphis, TN 38152, U.S.A.e-mail: [email protected]

Sorin G. GalDepartment of Mathematics and Computer ScienceUniversity of OradeaStr. Armatei Romane 5410087 Oradea, Romaniae-mail: [email protected]

Received: June 5, 2009.Revised: October 26, 2009.Accepted: November 9, 2009.

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Approximation by Complex Bernstein-Durrmeyer Polynomials in Compact Disks