Applied Mathematics and Computation 217 (2010) 1913–1920

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Approximation by complex genuine Durrmeyer typepolynomials in compact disks

Sorin G. GalDepartment of Mathematics and Computer Science, University of Oradea, Str. Universitatii No. 1, 410087 Oradea, Romania

a r t i c l e i n f o

Keywords:Complex genuine Durrmeyer polynomialsSimultaneous approximationVoronovskaja-type result

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.06.046

E-mail address: galso@uoradea.ro

a b s t r a c t

In this paper, the order of simultaneous approximation and Voronovskaja kind results withquantitative estimate for the complex genuine Durrmeyer polynomials attached to analyticfunctions on compact disks are obtained. In this way, we put in evidence the overconver-gence phenomenon for the genuine Durrmeyer polynomials, namely the extensions of theapproximation properties (with quantitative estimates) from real intervals to compactdisks in the complex plane.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

Concerning the convergence of Bernstein polynomials in the complex plane, Bernstein proved (see e.g. [16]) that iff : G! C is analytic in the open set G � C, with D1 � G (where D1 ¼ fz 2 C : jzj < 1g), then the complex Bernstein polyno-

mials Bnðf ÞðzÞ ¼Pn

k¼0nk

� �zkð1� zÞn�kf ðk=nÞ, uniformly converge to f in D1.

Exact estimates of order O(1/n) of this uniform convergence and, in addition, of the simultaneous approximation, we haveobtained in [4] and in the book [5]. A Voronovskaja-type result with quantitative estimate for the complex Bernstein poly-nomials in compact disks we have proved in [6].

Similar results for the complex Bernstein–Stancu and Kantorovich–Stancu polynomials we have obtained in [7–9] and forthe complex Favard–Szász–Mirakjan operators in [10]. All these results mentioned above, together with analogous results forother complex Bernstein type operators like those of Baskakov, Bernstein–Butzer and Balázs–Szabados can be found in thebook [11]. Also, note that analogous results for the complex classical Bernstein–Durrmeyer operators were obtained in [2].

The goal of this paper is to obtain approximation results for the complex genuine Durrmeyer polynomials (which are dif-ferent from the complex classical Bernstein–Durrmeyer operators), given explicitly and studied in the case of real variable ine.g. [1,3,13,18,19] and some of its q-generalizations in [14,15], defined by

Unðf ÞðzÞ ¼ f ð0Þpn;0ðzÞ þ f ð1Þpn;nðzÞ þ ðn� 1ÞXn�1

k¼1

pn;kðzÞZ 1

0pn�2;k�1ðtÞf ðtÞdt;

� �

where pn;kðzÞ ¼

nk zkð1� zÞn�k.

Note that all these results put in evidence the overconvergence phenomenon for the genuine Durrmeyer polynomials,that is the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks inthe complex plane.

. All rights reserved.

1914 S.G. Gal / Applied Mathematics and Computation 217 (2010) 1913–1920

2. Approximation results

First let us note that from the formula of definition it is easy to show that Un(e0)(z) = 1, Un(e1)(z) = e1(z),Unðe2ÞðzÞ ¼ e2ðzÞ þ 2zð1�zÞ

nþ1 . We use the notations epðzÞ ¼ zp; p 2 NSf0g; z 2 C.

The following recurrence formula will be very useful.

Lemma 2.1. For all p 2 NSf0g;n 2 N and z 2 C we have

Unðepþ1ÞðzÞ ¼zð1� zÞ

nþ pU0nðepÞðzÞ þ

nzþ pnþ p

UnðepÞðzÞ:

Proof. In the case of p = 0 the recurrence is immediate from Un(e0)(z) = 1 and Un(e1)(z) = e1(z). Therefore, let us suppose thatp P 1.

Denoting for simplicity

I ¼Z 1

0

n� 2k� 1

� �tkþp�1ð1� tÞn�k�1dt ¼

n� 2k� 1

� �Bðkþ p;n� kÞ;

from the formula of definition we can write

UnðepÞðzÞ ¼ zn þ ðn� 1ÞXn�1

k¼1

pn;kðzÞ � I;

which implies

U0nðepÞðzÞ ¼ nzn�1 þ ðn� 1ÞXn�1

k¼1

nk

� �zk�1ð1� zÞn�k � ½kI� � ðn� 1Þ

Xn�1

k¼1

nk

� �zkð1� zÞn�k�1 � ½ðn� kÞI�

¼ nzn�1 � n1� z

ðn� 1ÞXn�1

k¼1

pn;kðzÞ � I þ1

zð1� zÞ ðn� 1ÞXn�1

k¼1

pn;kðzÞ � ½kI�" #

¼ nzn

z� n

1� zUnðepÞðzÞ � zn� �

þ 1zð1� zÞ ðn� 1Þ

Xn�1

k¼1

pn;kðzÞ � ½kI�" #

¼ 1zð1� zÞ nzn þ ðn� 1Þ

Xn�1

k¼1

pn;kðzÞ � ½kI�" #

� n1� z

UnðepÞðzÞ:

But by the formula for the Euler’s function B(p,q) see e.g. [17, Exercise 1.31 b), p. 13], Bðpþ 1; qÞ ¼ ppþq Bðp; qÞ, for all

p; q 2 R with p, q > 0, we obtain

Bðkþ pþ 1;n� kÞ ¼ kþ pnþ p

Bðkþ p;n� kÞ;

which implies

kBðkþ p;n� kÞ ¼ ðnþ pÞBðkþ pþ 1;n� kÞ � pBðkþ p;n� kÞ:

Replacing above we obtain

U0nðepÞðzÞ ¼1

zð1� zÞ nzn þ ðn� 1ÞXn�1

k¼1

pn;kðzÞn� 2k� 1

� �½ðnþ pÞBðkþ pþ 1;n� kÞ � pBðkþ p;n� kÞ�

" #� n

1� zUnðepÞðzÞ

¼ 1zð1� zÞ ðnþ pÞUnðepþ1ÞðzÞ � pUnðepÞðzÞ

� �� n

1� zUnðepÞðzÞ;

which by multiplication with z(1 � z) becomes the recurrence in the statement. h

Also, the next lemma will be useful.

Lemma 2.2(i) For all n 2 N and p 2 N

Sf0g we have Un(ep)(1) = 1.

(ii) For all n; p 2 N and z 2 C we have

UnðepÞðzÞ ¼ðn� 1Þ!ðn� 1þ pÞ!

Xn

k¼0

n

k

� �Dk

1Fpð0Þzk ¼ ðn� 1Þ!ðn� 1þ pÞ!

Xminfn;pg

k¼0

n

k

� �Dk

1Fpð0Þzk;

� �

where FpðvÞ ¼ Pp�1

j¼0 ðv þ jÞ for all v � 0;Dk1Fpð0Þ ¼

Pkj¼0ð�1Þj k

jFpðk� jÞ and Dk

1Fpð0Þ � 0 for all k and p.

S.G. Gal / Applied Mathematics and Computation 217 (2010) 1913–1920 1915

Proof

(i) From the relationship (obtained in the proof of Lemma 2.1)

UnðepÞðzÞ ¼ zn þ ðn� 1ÞXn�1

k¼1

pn;kðzÞ � I;

it is immediate that Un(ep)(1) = 1.

(ii) Since Un(e0)(z) = 1 we can suppose that p � 1. We have

UnðepÞðzÞ ¼ zn þ ðn� 1ÞXn�1

k¼1

pn;kðzÞ � I;

where I ¼ n� 2k� 1

� �Bðkþ p;n� kÞ. Taking into account the formula Bðp; qÞ ¼ ðp�1Þ!ðq�1Þ!

ðpþq�1Þ! ; p; q 2 N, we easily obtain

I ¼ ðn� 2Þ!ðk� 1Þ!ðn� k� 1Þ! �

ðkþ p� 1Þ!ðn� k� 1Þ!ðn� 1þ pÞ! ¼ ðn� 2Þ!

ðn� 1þ pÞ! ½kðkþ 1Þ . . . ðkþ p� 1Þ� ¼ ðn� 2Þ!ðn� 1þ pÞ! FpðkÞ;

where FpðvÞ ¼ Pp�1j¼0 ðv þ jÞ. It is clear here that Fp(v) and its derivatives of any order are �0 for all v � 0, which imply

that Dk1Fpð0Þ � 0 for all k and p.

Therefore we can write

UnðepÞðzÞ ¼ zn þ ðn� 1Þ!ðn� 1þ pÞ!

Xn�1

k¼1

pn;kðzÞ � FpðkÞ;

and by simple reasonings we easily obtain

UnðepÞðzÞ ¼ zn þ ðn� 1Þ!ðn� 1þ pÞ!

Xn

k¼0

pn;kðzÞ � FpðkÞ � pn;0ðzÞFpð0Þ � pn;nðzÞFpðnÞ" #

¼ zn þ ðn� 1Þ!ðn� 1þ pÞ!

Xn

k¼0

nk

� �Dk

1Fpð0Þzk � zn½nðnþ 1Þ . . . ðn� 1þ pÞ�" #

¼ ðn� 1Þ!ðn� 1þ pÞ!

Xn

k¼0

nk

� �Dk

1Fpð0Þzk

¼ ðn� 1Þ!ðn� 1þ pÞ!

Xminfn;pg

k¼0

nk

� �Dk

1Fpð0Þzk;

which proves the lemma. h

The first main result is the following upper estimate.

Corollary 2.3. Let r P 1.

(i) For all p;n 2 NSf0g and jzj � r we have jUn(ep)(z)j � rp.

(ii) Let f ðzÞ ¼P1

k¼0ckzk for all jzj < R and take 1 � r < R. For all jzj � r and n 2 N we have

jUnðf ÞðzÞ � f ðzÞj � Crðf Þn

;

where Crðf Þ ¼ 2P1

p¼2jcpjpðp� 1Þrp <1.

Proof

(i) By Lemma 2.2, (i) and (ii) it is immediate that

ðn� 1Þ!ðn� 1þ pÞ!

Xn

k¼0

n

k

� �Dk

1Fpð0Þ ¼ 1;

which implies

jUnðepÞðzÞj �ðn� 1Þ!ðn� 1þ pÞ!

Xn

k¼0

n

k

� �Dk

1Fpð0Þrk � rp ðn� 1Þ!ðn� 1þ pÞ!

Xn

k¼0

n

k

� �Dk

1Fpð0Þ ¼ rp;

which proves (i).

1916 S.G. Gal / Applied Mathematics and Computation 217 (2010) 1913–1920

(ii) First we prove that Unðf ÞðzÞ ¼P1

k¼0ckUnðekÞðzÞ. Indeed, denoting fmðzÞ ¼Pm

j¼0cjzj; jzj � r;m 2 N, since from the linearityof Un we obviously have UnðfmÞðzÞ ¼

Pmk¼0ckUnðekÞðzÞ, it suffices to prove that for any fixed n 2 N and jzj � r with r P 1,

we have limm?1Un(fm)(z) = Un(f)(z). But this is immediate from limm?1kfm � fkr = 0 (here kfkr = maxjzj�r{jf(z)j}) andfrom the inequality

jUnðfmÞðzÞ � Unðf ÞðzÞj � jfmð0Þ � f ð0Þj � jð1� zÞnj þ jfmð1Þ � f ð1Þj � jznj þ ðn� 1ÞXn�1

k¼1

jpn;kðzÞj �Z 1

0pn�2;k�1ðtÞjfmðtÞ � f ðtÞjdt

� Cr;nkfm � fkr;

valid for all jzj � r, where simple calculation gives

Cr;n ¼ ð1þ rÞn þ rn þ ðn� 1ÞXn�1

k¼1

n

k

� �ð1þ rÞn�krk �

Z 1

0pn�2;k�1ðtÞdt:

Therefore we get

jUnðf ÞðzÞ � f ðzÞj �X1p¼0

jcpj � jUnðepÞðzÞ � epðzÞj ¼X1p¼1

jcpj � jUnðepÞðzÞ � epðzÞj;

since Un(e0)(z) = e0(z) = 1.We have two cases :(1) 1 � p � n ; 2) p > n.

(Case 1). From Lemma 2.2 (i) and (ii) we obtain

UnðepÞðzÞ � epðzÞ ¼ zp ðn� 1Þ!ðn� 1þ pÞ!

n

p

� �Dp

1Fpð0Þ � 1� �

þ ðn� 1Þ!ðn� 1þ pÞ!

Xp�1

k¼0

n

k

� �Dk

1Fpð0Þzk;

and

jUnðepÞðzÞ � epðzÞj � rp 1� ðn� 1Þ!ðn� 1þ pÞ!

np

� �Dp

1Fpð0Þ� �

þ rp 1� ðn� 1Þ!ðn� 1þ pÞ!

np

� �Dp

1Fpð0Þ� �

� 2rp 1� ðn� 1Þ!ðn� 1þ pÞ!

np

� �Dp

1Fpð0Þ� �

:

Here it is easy to see that we can write

ðn� 1Þ!ðn� 1þ pÞ!

n

p

� �Dp

1Fpð0Þ ¼ðn� 1Þ!ðn� 1þ pÞ!

n

p

� �p! ¼ Pp

j¼1nþ j� pnþ j� 1

:

But applying the formula (easily proved by mathematical induction)

1�Pkj¼1xj �

Xk

j¼1

ð1� xjÞ; 0 � xj � 1; j ¼ 1; . . . ; k;

for xj ¼ nþj�pnþj�1 and k = p, we obtain

1�Ppj¼1

nþ j� pnþ j� 1

�Xp

j¼1

1� nþ j� pnþ j� 1

� �¼ ðp� 1Þ

Xp

j¼1

1nþ j� 1

� ðp� 1Þpn

:

Therefore it follows

jUnðepÞðzÞ � epðzÞj �2pðp� 1Þrp

n:

(Case 2). By (i) and by p > n P 1 we obtain

jUnðepÞðzÞ � epðzÞj � jUnðepÞðzÞj þ jepðzÞj � 2rp <2pn

rp � 2pðp� 1Þn

rp:

In conclusion, from both cases 1) and 2) we obtain for all p;n 2 N

jUnðepÞðzÞ � epðzÞj �2pðp� 1Þrp

n;

which implies

jUnðf ÞðzÞ � f ðzÞj � 2n

X1p¼1

jcpjpðp� 1Þrp

S.G. Gal / Applied Mathematics and Computation 217 (2010) 1913–1920 1917

and proves the corollary. h

The Voronovskaja’s theorem for the real case with a quantitative estimate is obtained by Proposition 7.4 in [12] in thefollowing form:

Unðf ÞðxÞ � f ðxÞ � xð1� xÞnþ 1

f 00ðxÞ����

���� � xð1� xÞnþ 1

x1 f 002

3ffiffiffiffiffiffiffiffiffiffiffiffinþ 3p

� �;

n 2 N; x 2 ½0;1�.For the complex genuine Durrmeyer polynomials it is expected to derive a formula of the form

Unðf ÞðzÞ � f ðzÞ � zð1� zÞnþ 1

f 00ðzÞ����

���� � Mr;f

n2 ;

for all n 2 N; jzj � r.Indeed, in what follows we will prove the Voronovskaja theorem with a quantitative estimate for the complex version of

genuine Durrmeyer polynomials, as follows.

Theorem 2.4. Let R > 1 and suppose that f : DR ! C is analytic in DR ¼ fz 2 C; jzj < Rg, that is we can write f ðzÞ ¼P1

k¼0ckzk, forall z 2 DR. For any fixed r 2 [1,R) and for all n 2 N; jzj � r, the following Voronovskaja-type result holds

Unðf ÞðzÞ � f ðzÞ � zð1� zÞf 00ðzÞnþ 1

�������� � Mrðf Þ

n2 ;

where Mrðf Þ ¼ 16P1

k¼3jckjkðk� 1Þðk� 2Þ2rk <1.

Proof. Denoting pk,n(z) = Un(ek)(z), by the proof of Corollary 2.3 (ii), we can write Unðf ÞðzÞ ¼P1

k¼0ckpk;nðzÞ. Also, since

zð1� zÞf 00ðzÞnþ 1

¼ zð1� zÞnþ 1

X1k¼2

kðk� 1Þckzk�2;

taking into account that Un(e0)(z) = 1,Un(e1)(z) = e1(z) we immediately obtain

Unðf ÞðzÞ � f ðzÞ � zð1� zÞf 00ðzÞnþ 1

�������� �X1

k¼2

jckj � pk;nðzÞ � ekðzÞ �kðk� 1Þð1� zÞzk�1

nþ 1

��������;

for all z 2 DR;n 2 N.In what follows, we will use the recurrence obtained in Lemma 2.1:

pkþ1;nðzÞ ¼zð1� zÞ

nþ kp0k;nðzÞ þ

nzþ knþ k

pk;nðzÞ;

for all n 2 N; z 2 C and k = 0,1, . . ..If we denote

Ek;nðzÞ ¼ pk;nðzÞ � ekðzÞ �kðk� 1Þð1� zÞzk�1

nþ 1;

then it is clear that Ek,n(z) is a polynomial of degree � k and by a simple calculation and the use of the above recurrence weobtain the following relationship

Ek;nðzÞ ¼zð1� zÞ

nþ k� 1E0k�1;nðzÞ þ

nzþ k� 1nþ k� 1

Ek�1;nðzÞ þzk�2ð1� zÞðk� 1Þðk� 2Þðnþ 1Þðnþ k� 1Þ ½ð2k� 3Þ � 2kz�;

valid for all k P 2;n 2 N and jzj � r.For all k;n 2 N; k P 2 and jzj � r, it implies

jEk;nðzÞj �rð1þ rÞ

nþ k� 1jE0k�1;nðzÞj þ

nr þ k� 1nþ k� 1

jEk�1;nðzÞj þrk�2ð1þ rÞðk� 1Þðk� 2Þðnþ 1Þðnþ k� 1Þ ½2k� 3þ 2kr�:

Since rð1þrÞnþkþ1 �

rð1þrÞn and nrþk�1

nþk�1 � r it follows

jEk;nðzÞj �rð1þ rÞ

njE0k�1;nðzÞj þ rjEk�1;nðzÞj þ

rk�2ð1þ rÞ22kðk� 1Þðk� 2Þn2 :

Now we will estimate jE0k�1;nðzÞj, for k P 2. We will use the estimate obtained in the proof of Corollary 2.3 (ii),

jpk;nðzÞ � ekðzÞj �2kðk� 1Þrk

n;

for all k;n 2 N; jzj � r, with 1 � r.

1918 S.G. Gal / Applied Mathematics and Computation 217 (2010) 1913–1920

Taking into account that Ek�1,n(z) is a polynomial of degree � (k � 1), by the well-known Bernstein’s inequality we obtain

jE0k�1;nðzÞj �k� 1

rkEk�1;nkr �

k� 1r

kpk�1;n � ek�1kr þðk� 1Þðk� 2Þek�2ð1� e1Þ

n

r

� �

� k� 1r

2ðk� 1Þðk� 2Þrk�1

nþ rk�2ðr þ 1Þðk� 1Þðk� 2Þ

n

� �� k� 1

r� ðk� 1Þðk� 2Þrk�2

n2r þ ðr þ 1Þ½ �

� 4ðk� 1Þ2ðk� 2Þrk�2

n;

for all k P 2 and jzj � r.This implies

rð1þ rÞn

jE0k�1;nðzÞj �4ðk� 1Þ2ðk� 2Þð1þ rÞrk�1

n2 ;

and

jEk;nðzÞj � rjEk�1;nðzÞj þ4ðk� 1Þ2ðk� 2Þð1þ rÞrk�1

n2 þ rk�2ð1þ rÞ22kðk� 1Þðk� 2Þn2

� rjEk�1;nðzÞj þ8ðk� 1Þ2ðk� 2Þrk

n2 þ 8rkkðk� 1Þðk� 2Þn2 � rjEk�1;nðzÞj þ

16kðk� 1Þðk� 2Þrk

n2 :

But E0,n(z) = E1,n(z) = E2,n(z) = 0, for any z 2 C and therefore by writing the last inequality for k = 3,4, . . . , we easily obtain,step by step the following

jEk;nðzÞj �16rk

n2

Xk

j¼3

jðj� 1Þðj� 2Þ" #

� 16rkkðk� 1Þðk� 2Þ2

n2 :

As a conclusion, we obtain

Unðf ÞðzÞ � f ðzÞ � zð1� zÞf 00ðzÞnþ 1

�������� �X1

k¼3

jckj � jEk;nðzÞj �16n2

X1k¼3

jckjkðk� 1Þðk� 2Þ2rk:

Note that since f ð4ÞðzÞ ¼P1

k¼5ckkðk� 1Þðk� 2Þðk� 3Þzk�4, and the series is absolutely convergent in jzj � r, it easily followsthat

P1k¼5jckjkðk� 1Þðk� 2Þðk� 3Þrk�4 <1, which immediately implies that

P1k¼3jckjkðk� 1Þðk� 2Þ2rk <1 and proves the

theorem. h

By using the above Voronovskaja’s theorem, in what follows we will obtain the exact order in approximation by the com-plex genuine Durrmeyer polynomials and their derivatives. In this sense we present the following results.

Theorem 2.5. Let R > 1;DR ¼ fz 2 C; jzj < Rg and let us suppose that f : DR ! C is analytic in DR, that is we can writef ðzÞ ¼

P1k¼0ckzk, for all z 2 DR. If f is not a polynomial of degree � 1, then for any r 2 [1,R) we have

kUnðf Þ � fkr PCrðf Þnþ 1

; n 2 N;

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

Proof. For all z 2 DR and n 2 N we have

Unðf ÞðzÞ � f ðzÞ ¼ 1nþ 1

zð1� zÞf 00ðzÞ þ 1nþ 1

ðnþ 1Þ2 Unðf ÞðzÞ � f ðzÞ � zð1� zÞf 00ðzÞnþ 1

� �� �� �:

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

kF þ Gkr P j kFkr � kGkr jP kFkr � kGkr:

It follows

kUnðf Þ � fkr P1

nþ 1e1ð1� e1Þf 00k kr �

1nþ 1

ðnþ 1Þ2 Unðf Þ � f � e1ð1� e1Þf 00nþ 1

r

� �� �:

Taking into account that by hypothesis f is not a polynomial of degree � 1 in DR, we get ke1(1 � e1)f00kr > 0. Indeed,supposing the contrary it follows that z(1 � z)f00(z) = 0 for all z 2 Dr , which by the analyticity of f in the disk jzj � r (withr P 1) clearly implies f00(z) = 0 for all jzj � r, that is f is a polynomial of degree � 1 for all jzj � r, a contradiction.

But by Theorem 2.4 we immediately get

ðnþ 1Þ2 Unðf Þ � f � e1ð1� e1Þf 00nþ 1

r

� 4Mrðf Þ:

S.G. Gal / Applied Mathematics and Computation 217 (2010) 1913–1920 1919

Therefore, there exists an index n0 depending only on f and r, such that for all n P n0 we have

e1ð1� e1Þf 00k kr �1

nþ 1ðnþ 1Þ2 Unðf Þ � f � e1ð1� e1Þf 00

nþ 1

r

� �P

12

e1ð1� e1Þf 00k kr ;

which immediately implies

kUnðf Þ � fkr P1

nþ 1� 12

e1ð1� e1Þf 00k kr ; 8 n P n0:

For n 2 {1, . . .,n0 � 1} we obviously have kUnðf Þ � fkr P Mr;nðf Þnþ1 with Mr,n(f) = (n + 1) � kUn(f) � fkr > 0. Indeed, if we would

have that kUn(f) � fkr = 0, then would follow Un(f)(z) = f(z) for all jzj � r, which is valid only for f a polynomial of degree�1, contradicting the hypothesis on f in the statement.

Therefore, finally we get kUnðf Þ � fkr P Crðf Þnþ1 for all n, where

Crðf Þ ¼min Mr;1ðf Þ; . . . ;Mr;n0�1ðf Þ;12

e1ð1� e1Þf 00k kr

� �:

This completes the proof. h

Combining now Theorem 2.5 with Corollary 2.3 (ii) we immediately get the following.

Corollary 2.6. Let R > 1;DR ¼ fz 2 C; jzj < Rg and let us suppose that f : DR ! C is analytic in DR. If f is not a polynomial ofdegree �1, then for any r 2 [1,R) we have

kUnðf Þ � fkr �1n;n 2 N;

where the constants in the equivalence depend only on f and r.For the derivatives of complex genuine Durrmeyer polynomials we can state the following result.

Theorem 2.7. Let DR ¼ fz 2 C; jzj < Rg be with R > 1 and let us suppose that f : DR ! C is analytic in DR, i.e. f ðzÞ ¼P1

k¼0ckzk, forall z 2 DR. Also, let 1 � r < r1 < R and p 2 N be fixed. If f is not a polynomial of degree �max{1,p � 1}, then we have

kUðpÞn ðf Þ � f ðpÞkr �1n;

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

Proof. Denoting by C the circle of radius r1> and center 0 (where r1 > r P 1), by the Cauchy’s formulas it follows that for alljzj � r and n 2 N we have

UðpÞn ðf ÞðzÞ � f ðpÞðzÞ ¼ p!

2pi

ZC

Unðf ÞðvÞ � f ðvÞðv � zÞpþ1 dv;

which by Corollary 2.3 (ii) and by the inequality jv � zjP r1 � r valid for all jzj � r and v 2 C, immediately implies

kUðpÞn ðf Þ � f ðpÞkr �p!

2p� 2pr1

ðr1 � rÞpþ1 kUnðf Þ � fkr1� Cr1 ðf Þ

p!r1

nðr1 � rÞpþ1 :

It remains to prove the lower estimate for kUðpÞn ðf Þ � f ðpÞkr .For this purpose, as in the proof of Theorem 2.5, for all v 2 C and n 2 N we have

Unðf ÞðvÞ � f ðvÞ ¼ 1nþ 1

vð1� vÞf 00ðvÞ þ 1nþ 1

ðnþ 1Þ2 Unðf ÞðvÞ � f ðvÞ � vð1� vÞf 00ðvÞnþ 1

� �� �� �;

which replaced in the above Cauchy’s formula implies

UðpÞn ðf ÞðzÞ � f ðpÞðzÞ ¼ 1nþ 1

p!

2pi

ZC

vð1� vÞf 00ðvÞðv � zÞpþ1 dv þ 1

nþ 1� p!

2pi

ZC

ðnþ 1Þ2 Unðf ÞðvÞ � f ðvÞ � vð1�vÞf 00 ðvÞnþ1

�ðv � zÞpþ1 dv

8<:

9=;

¼ 1nþ 1

zð1� zÞf 00ðzÞ½ �ðpÞ þ 1nþ 1

� p!

2pi

ZC

ðnþ 1Þ2 Unðf ÞðvÞ � f ðvÞ � vð1�vÞf 00 ðvÞnþ1

�ðv � zÞpþ1 dv

8<:

9=;:

Passing now to k � kr it follows

1920 S.G. Gal / Applied Mathematics and Computation 217 (2010) 1913–1920

kUðpÞn ðf Þ � f ðpÞkr P1

nþ 1e1ð1� e1Þf 00½ �ðpÞ

r� 1

nþ 1p!

2p

ZC

ðnþ 1Þ2 Unðf ÞðvÞ � f ðvÞ � vð1�vÞf 00ðvÞnþ1

�ðv � zÞpþ1 dv

r

8<:

9=;;

where by using Theorem 2.4 we get

p!

2p

ZC

ðnþ 1Þ2 Unðf ÞðvÞ � f ðvÞ � vð1�vÞf 00 ðvÞnþ1

�ðv � zÞpþ1 dv

r

� p!

2p� 2pr1ðnþ 1Þ2

ðr1 � rÞpþ1 Unðf Þ � f � e1ð1� e1Þf 00nþ 1

r1

� 4Mr1 ðf Þp!r1

ðr1 � rÞpþ1 :

But by hypothesis on f we have k[e1(1 � e1)f00](p) kr > 0.Indeed, supposing the contrary it follows that z(1 � z)f00(z) is a polynomial of degree � p � 1.Now, if p = 1 then we get z(1 � z)f00(z) = C, which implies f 00ðzÞ ¼ C

zð1�zÞ, for all jzj � r with r P 1. But since f00(z) is analyticin jzj � r, this necessarily implies C = 0, that is f(z) is a polynomial of degree � 1 = max{1,p � 1}, a contradiction with thehypothesis on f.

For p = 2 we get z(1 � z)f00(z) = Az + B, which implies f 00ðzÞ ¼ AzþBzð1�zÞ, for all jzj � r. But since f00(z) is analytic in jzj � r, this

necessarily implies A = B = 0 (because contrariwise f would have a pole at z = 0 or at z = 1), and therefore f would be apolynomial of degree 6 1 = max{1,p � 1}, a contradiction.

If p P 3 then we get z(1 � z)f00(z) = Qp�1(z), where Qp�1(z) is a polynomial of degree 6 p � 1. This implies f 00ðzÞ ¼ Qp�1ðzÞzð1�zÞ , for

all jzj 6 r (with r P 1). Then the analyticity of f obviously implies that Qp�1(z) = z(1 � z)Rp�3(z) where Rp�3(z) is a polynomialof degree � p � 3 (because contrariwise f would have a pole at z = 0 or at z = 1). Therefore, necessarily we get f00(z) = Rp�3(z),that is f(z) is a polynomial of degree � p � 1 = max{1,p � 1}, which again contradicts the hypothesis on f.

In continuation reasoning exactly as in the proof of Theorem 2.5, we immediately get the desired conclusion. h

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