Applied Mathematics and Computation 234 (2014) 309–315

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Applied Mathematics and Computation

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

Approximation by complex Phillips–Stancu operatorsin compact disks under exponential growth conditions

http://dx.doi.org/10.1016/j.amc.2014.02.0490096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail addresses: galso@uoradea.ro (S.G. Gal), vijaygupta2001@hotmail.com (V. Gupta).

Sorin G. Gal a, Vijay Gupta b,⇑a Department of Mathematics and Computer Science, University of Oradea, Str. Universitatii No. 1, 410087 Oradea, Romaniab Department of Mathematics, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi 110078, India

a r t i c l e i n f o a b s t r a c t

Keywords:Complex Phillips–Stancu operatorsVoronovskaja kind asymptotic formulaExact order of approximation in compactdisksSimultaneous approximation

The present article deals with the approximation properties of the complex Phillips–Stancuoperators. Here we study a Voronovskaja kind asymptotic formula with quantitative esti-mates for these operators attached to analytic functions and having growth of exponentialorder on compact disks. Also, we obtain the exact order of approximation.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Almost six decades ago, while finding the integral modification of Szász basis function sn;vðxÞ ¼ e�nx ðnxÞvv !

, Phillips [12]introduced the following operators and studied its approximation properties in real domain

Sða;bÞn ðf ÞðxÞ ¼ nX1v¼1

sn;vðxÞZ þ1

0sn;v�1ðtÞf ðtÞdt þ e�nxf ð0Þ;

Finta and Gupta [2] obtained direct and converse results for the Phillips operators. Very recently Heilmann and Tachev [11]also studied some interesting properties of these operators. About ten years ago Srivastava and Guppta [13] considered ageneral sequence of linear positive operators, which was later termed as Srivastava–Gupta operators in [1,14]. The specialcase of Srivastava–Gupta operators include Phillips operators. It is observed that the other case which include Baskakov–Durrmeyer type operators are not bounded. So here we are dealing with the Phillips operators.

In [3] Gal presented the overconvergence properties of certain operators in complex domain. After that many papers ondifferent operators appeared in literature see e.g. [4–7,9,10] etc. In this direction, very recently we [8] established approx-imation properties of the Phillips operators in complex domain. For 0 6 a 6 b the Stancu variant of the complex Phillipsoperator can be defined as

Sða;bÞn ðf ÞðzÞ ¼ nX1v¼1

sn;vðzÞZ þ1

0sn;v�1ðtÞf

nt þ anþ b

� �dt þ e�nzf

anþ b

� �;

where sn;vðzÞ ¼ e�nz ðnzÞvv !

.In the present paper, we estimate the Voronovskaja kind asymptotic formula with quantitative estimates for the

Phillips–Stancu operators attached to analytic functions having exponential growth on compact disks. We also establishthe exact order of approximation for these operators.

310 S.G. Gal, V. Gupta / Applied Mathematics and Computation 234 (2014) 309–315

Throughout the present article we consider DR ¼ fz 2 C : jzj < Rg. By HR, we mean the class of all functions satisfying:f : ½R;þ1Þ [DR ! C is continuous in ½R;þ1Þ [DR, analytic in DR i.e. f ðzÞ ¼

P1k¼0ckzk, for all z 2 DR.

2. Lemmas

In the sequel, we need the following basic lemmas:

Lemma 1. Let 0 6 a 6 b and suppose that f : ½R;þ1ÞS

DR is analytic in DR and there exists B;C > 0 such that jf ðxÞj 6 CeBx, for allx 2 ½R;þ1Þ. Denoting f ðzÞ ¼

P1k¼0ckzk; z 2 DR, we have Sða;bÞn ðf ÞðzÞ ¼

P1k¼0ckSða;bÞn ðekÞðzÞ, for all z 2 DR and n > B� b.

Proof. For any m 2 N and 0 < r < R, let us define

fmðzÞ ¼Xm

j¼0

cjzj if jzj 6 r and f mðxÞ ¼ f ðxÞ if x 2 ðr;þ1Þ:

Since jfmðzÞj 6P1

j¼0jcjj � rj :¼ Cr , for all jzj 6 r and m 2 N; f is continuous on ½r;R�, from the hypothesis on f it is clear that forany m 2 N it follows jfmðxÞj 6 Cr;ReBx, for all x 2 ½0;þ1Þ. This implies that for each fixed m;n 2 N; n > B� b and z,

jSða;bÞn ðfmÞðzÞj 6 Cr;Rje�nzjeBa=ðnþbÞX1j¼1

ðnjzjÞj

j!nZ 1

0� nj�1

ðj� 1Þ! tj�1etðBn=ðnþbÞ�nÞdt þ jc0j !

¼ Cr;Rje�nzj � ðjc0j þ 1ÞeBa=ðnþbÞX1j¼0

ðnjzjÞj

j!� nj

ðn� Bn=ðnþ bÞÞj<1;

since by the ratio criterium the last series is convergent. Therefore Sða;bÞn ðfmÞðzÞ is well-defined.Denoting

fm;kðzÞ ¼ ckekðzÞ if jzj 6 r and f m;kðxÞ ¼f ðxÞ

mþ 1if x 2 ðr;1Þ;

it is clear that each fm;k is of exponential growth on ½0;1Þ and that fmðzÞ ¼Pm

k¼0fm;kðzÞ. Since from the linearity of Sða;bÞn wehave

Sða;bÞn ðfmÞðzÞ ¼Xm

k¼0

ckSða;bÞn ðekÞðzÞ; for all jzj 6 r;

it suffices to prove that limm!1Sða;bÞn ðfmÞðzÞ ¼ Sða;bÞn ðf ÞðzÞ for any fixed n 2 N and jzj 6 r. But this is immediate fromlimm!1kfm � fkr ¼ 0, from kfm � fkB½0;þ1Þ 6 kfm � fkr and from the inequality

jSða;bÞn ðfmÞðzÞ � Sða;bÞn ðf ÞðzÞj 6 je�nzj � enjzj � kfm � fkB½0;1Þ 6 Mr;nkfm � fkr;

valid for all jzj 6 r. Here k � kB½0;þ1Þ denotes the uniform norm on C½0;þ1Þ-the space of all complex-valued bounded functionson ½0;þ1Þ. h

Lemma 2. If we denote SnðekÞðzÞ ¼: Sð0;0Þn ðekÞðzÞ, where ekðzÞ ¼ zk, then for all jzj 6 r with r P 1;n 2 N and k ¼ 0;1; . . . ;2 . . ., wehave the estimate jSnðekÞðzÞj 6 ð2kÞ! � rk.

Proof. We will use the recurrence formula as in Lemma 2 in [8]:

Snðekþ1ÞðzÞ ¼zn

S0nðekÞðzÞ þnzþ k

nSnðekÞðzÞ; Snðe0ÞðzÞ ¼ 1:

For k ¼ 0, we get jSnðe1ÞðzÞj 6 r, for all jzj 6 r;n 2 N. For k ¼ 1, we get

jSnðe2ÞðzÞj 6rn� kS0nðe1Þkr þ ðr þ 1=nÞr 6 rðr þ 2=nÞ:

In general, taking into account that SnðekÞðzÞ is a polynomial of degree k and that by the Bernstein inequality we havekS0nðekÞðzÞkr 6

kr � kSnðekÞkr , by mathematical induction we easily arrive at the inequality

jSnðekÞðzÞj 6Yk

j¼1

r þ 2j� 2n

� �¼ rk

Yk

j¼1

ð1þ ð2j� 2Þ=ðnrÞÞ 6 rkð2kÞ!;

for all jzj 6 r and k;n 2 N. h

S.G. Gal, V. Gupta / Applied Mathematics and Computation 234 (2014) 309–315 311

Lemma 3. Let 0 6 a 6 b, then

Sða;bÞn ðekÞðzÞ ¼Xk

j¼0

kj

� �njak�j

ðnþ bÞkSnðejÞðzÞ;

where Sn denotes Sð0;0Þn .

Proof. It is immediate. h

Theorem 1. Let 0 6 a 6 b; f 2 HR; 1 < R < þ1 and suppose that there exist M > 0 and A 2 ð1R ;1Þ, with the property thatjckj 6 M Ak

ð2kÞ!, for all k ¼ 0;1; . . ., (which implies jf ðzÞj 6 MeAjzj for all z 2 DR) and jf ðxÞj 6 CeBx, for all x 2 ½R;þ1Þ.

(i) Let 1 6 r < 1A. Then for all jzj 6 r and n 2 N with n > B� b, we have

jSða;bÞn ðf ÞðzÞ � f ðzÞj 6 Cr;A �nðbþ 1Þ þ b

nðnþ bÞ ;

where Cr;A ¼ 2M �P1

k¼1ðrAÞk <1;(ii) If 1 6 r < r1 <

1A are arbitrary fixed, then for all jzj 6 r and n; p 2 N with n > B� b,

j½Sða;bÞn �ðpÞðf ÞðzÞ � f ðpÞðzÞj 6 p!r1Cr1 ;A

ðr1 � rÞpþ1 �nðbþ 1Þ þ b

nðnþ bÞ ;

where Cr1 ;A is given as at the above point (i).

Proof.

(i) By Lemma 3, we get

Sða;bÞn ðekÞðzÞ � ekðzÞ ¼Xk�1

j¼0

kj

� �njak�j

ðnþ bÞkðSnðejÞðzÞ � ejðzÞÞ þ

Xk�1

j¼0

kj

� �njak�j

ðnþ bÞkejðzÞ þ

nk

ðnþ bÞkSnðekÞðzÞ � ekðzÞ;

which by using the estimate for kSnðejÞ � ejkr 6ð2jÞ!

n rj�1 in the proof of Theorem 16, (i) in [8], implies

kSða;bÞn ðekÞ � ekkr 6Xk�1

j¼0

kj

� �njak�j

ðnþ bÞkkSnðejÞ � ejkr þ

Xk�1

j¼0

kj

� �njak�j

ðnþ bÞkrj þ nk

ðnþ bÞkkSnðekÞ � ekkr þ 1� nk

ðnþ bÞk

!rk

6ðnþ aÞk

ðnþ bÞk� ð2kÞ!

nrk�1 þ rk ðnþ aÞk

ðnþ bÞk� nk

ðnþ bÞk

" #þ nk

ðnþ bÞk� ð2kÞ!

nrk�1 þ 1� nk

ðnþ bÞk

!rk

6ð2kÞ!

nrk�1 þ 2rk 1� nk

ðnþ bÞk

" #6 2 � ð2kÞ!

nrk�1 þ 2rk � kb

nþ b6 2

nðbþ 1Þ þ bnðnþ bÞ � ð2kÞ!rk:

Above we get the estimate 1� nk

ðnþbÞk6

kbnþb by using the inequality 1�

Qkj¼1xj 6

Pkj¼1ð1� xjÞ, valid for all 0 6 xj 6 1;

j ¼ 1; . . . ; k.In conclusion, by Lemma 1 we can write Sða;bÞn ðf ÞðzÞ ¼

P1k¼0ckSða;bÞn ðekÞðzÞ for all z 2 DR; n > B� b, which from the hypothesis

on ck immediately implies for all jzj 6 r

jSða;bÞn ðf ÞðzÞ � f ðzÞj 6X1k¼1

jckj � jSða;bÞn ðekÞðzÞ � ekðzÞj 6X1k¼1

2MAk

ð2kÞ!nðbþ 1Þ þ b

nðnþ bÞ � ð2kÞ!rk ¼ nðbþ 1Þ þ bnðnþ bÞ � 2M

X1k¼1

ðrAÞk

¼ Cr;A �nðbþ 1Þ þ b

nðnþ bÞ

where Cr;A ¼ 2M �P1

k¼1ðrAÞk <1 for all 1 6 r < 1A, taking into account that the series

P1k¼1uk is uniformly convergent in any

compact disk included in the open unit disk.(ii) Denoting by c the circle of radius r1 > r and center 0, since for any jzj 6 r and v 2 c, we have jv � zjP r1 � r, by the

Cauchy’s formulas it follows that for all jzj 6 r and n 2 N with n > B� b, we have

j½Sða;bÞn �ðpÞðf ÞðzÞ � f ðpÞðzÞj ¼ p!

2p

Zc

Sða;bÞn ðf ÞðvÞ � f ðvÞðv � zÞpþ1 dv

���������� 6 Cr1 ;A

nðbþ 1Þ þ bnðnþ bÞ �

p!

2p� 2pr1

ðr1 � rÞpþ1 ;

312 S.G. Gal, V. Gupta / Applied Mathematics and Computation 234 (2014) 309–315

which proves (ii) and the theorem.

h

The following Voronovskaja type result holds.

Theorem 2. Let f 2 HR;2 < R < þ1 and that there exist M > 0 and A 2 ð1R ;1Þ, with the property that jckj 6 M Ak

ð2kÞ!, for allk ¼ 0;1; . . ., (which implies jf ðzÞj 6 MeAjzj for all z 2 DR) and jf ðxÞj 6 CeBx, for all x 2 ½R;þ1Þ.

If 1 6 r < r þ 1 < 1A then there exists a constant Ca;b;r > 0 (depending only on a; b and r), such that for all jzj 6 r and n 2 N with

n > max Ar1�Ar ;B� �

, we have

Sða;bÞn ðf ÞðzÞ � f ðzÞ � a� bzn

f 0ðzÞ � zn

f 00ðzÞ����

���� 6 Cr;A;Mðf Þn2 þ MCa;b;r

ðnþ bÞ2�X1k¼0

kðk� 1ÞðArÞk;

where Cr;A;Mðf Þ ¼ 2Mrð1�ArÞ þ 4M

ðrþ1Þ2P1

k¼2ðk� 1Þ½Aðr þ 1Þ�k <1.

Proof. For all f 2 DR, let us consider

Sða;bÞn ðf ÞðzÞ � f ðzÞ � a� bzn

f 0ðzÞ � zn

f 00ðzÞ ¼ Snðf ÞðzÞ � f ðzÞ � zn

f 00ðzÞ þ Sða;bÞn ðf ÞðzÞ � Snðf ÞðzÞ �a� bz

nf 0ðzÞ:

Taking f ðzÞ ¼P1

k¼0ckzk, we get

Sða;bÞn ðf ÞðzÞ � f ðzÞ � a� bzn

f 0ðzÞ � zn

f 00ðzÞ ¼X1k¼0

ck SnðekÞðzÞ � zk � kðk� 1Þzk�1

n

� �

þX1k¼0

ck Sða;bÞn ðekÞðzÞ � SnðekÞðzÞ �a� bz

nkzk�1

� �:

By Theorem 17 of [8], for all n > maxfAr=ð1� ArÞ;Bg and jzj 6 r we have

X1k¼0

ck SnðekÞðzÞ � zk � kðk� 1Þzk�1

n

� ����������� 6 Cr;A;Mðf Þ

n2 ;

where Cr;A;Mðf Þ ¼ 2Mrð1�ArÞ þ 4M

ðrþ1Þ2P1

k¼2ðk� 1Þ½Aðr þ 1Þ�k <1.

To estimate the second term above, we use Lemma 3 and we rewrite it as follows

S a;bð Þn ðekÞðzÞ � SnðekÞðzÞ �

a� bzn

kzk�1 ¼Xk�1

j¼0

k

j

� �njak�j

nþ bð ÞkSnðejÞðzÞ þ

nk

nþ bð Þk� 1

!SnðekÞðzÞ �

a� bzn

kzk�1

¼Xk�2

j¼0

k

j

� �njak�j

nþ bð ÞkSnðejÞðzÞ þ

knk�1anþ bð Þk

Snðek�1ÞðzÞ �Xk�1

j¼0

k

j

� �njbk�j

nþ bð ÞkSnðekÞðzÞ �

a� bzn

kzk�1

¼Xk�2

j¼0

k

j

� �njak�j

nþ bð ÞkSnðejÞðzÞ þ

knk�1anþ bð Þk

Sn ek�1; zð Þ � zk�1� ��Xk�2

j¼0

k

j

� �njbk�j

nþ bð ÞkSnðekÞðzÞ þ

nk�1

nþ bð Þk�1 � 1

!a

nþ bkzk�1

þ knk�1b

nþ bð Þkzk � SnðekÞðzÞ� �

þ 1� nk�1

nþ bð Þk�1

!bz

nþ bkzk�1 � bða� bzÞ

nðnþ bÞ kzk�1:

By using Lemma 2 and the inequality

1� nk

nþ bð Þk6

Xk

j¼1

1� nnþ b

� �¼ kb

nþ b;

we get

Xk�2

j¼0

k

j

� �njak�j

nþ bð ÞkSnðejÞðzÞ

���������� 6

Xk�2

j¼0

k

j

� �njak�j

nþ bð ÞkSnðejÞðzÞ�� �� ¼Xk�2

j¼0

k� 1ð Þkk� j� 1ð Þ k� jð Þ

k� 2j

� �njak�j

nþ bð ÞkSnðejÞðzÞ�� ��

6k k� 1ð Þ

2a2

nþ bð Þ2rk�2ð2ðk� 2ÞÞ!

Xk�2

j¼0

njak�2�j

nþ bð Þk�2 6k k� 1ð Þ

2a2

nþ bð Þ2rk�2ð2ðk� 2ÞÞ!:

Therefore, taking into account the inequality jSnðekÞðzÞ � zkj 6 ð2kÞ!n rk�1, for all k; n 2 N, jzj 6 r in the proof of Theorem 1 in [8],

it easily follows that

S.G. Gal, V. Gupta / Applied Mathematics and Computation 234 (2014) 309–315 313

S a;bð Þn ðekÞðzÞ � SnðekÞðzÞ �

a� bzn

kzk�1����

���� 6 k k� 1ð Þa2

2 nþ bð Þ2rk�2ð2ðk� 2ÞÞ!þ a

2ðnþ bÞ2kð2ðk� 1ÞÞ!rk�2 þ k k� 1ð Þb2

2 nþ bð Þ2rkð2kÞ!

þ k k� 1ð Þab

nþ bð Þ2rk�1 þ kb

2ðnþ bÞ2ð2kÞ!rk�1 þ k k� 1ð Þb

nþ bð Þ2rk þ bð1þ rbÞkrk�1

nðnþ bÞ

and that there exists a constant Ca;b;r > 0 (depending only on a; b; r, which could be explicitly found by some calculation, butfor simplicity we do not make it here), such that

S a;bð Þn ðekÞðzÞ � SnðekÞðzÞ �

a� bzn

kzk�1����

���� 6 Ca;b;r

ðnþ bÞ2� kðk� 1Þrkð2kÞ!:

In conclusion,

X1k¼0

ck Sða;bÞn ðekÞðzÞ � SnðekÞðzÞ �a� bz

nkzk�1

� ����������� 6

X1k¼0

jckj � Sða;bÞn ðekÞðzÞ � SnðekÞðzÞ �a� bz

nkzk�1

��������

6MCa;b;r

ðnþ bÞ2�X1k¼0

kðk� 1ÞðArÞk;

which finally leads to the estimate in the statement. h

The following exact order of approximation can be obtained.

Theorem 3.

(i) In the hypothesis of Theorem 2, if f is not a polynomial of degree 6 0 then for all 1 6 r < r þ 1 < R we have

kSða;bÞn ðf Þ � fkr �1n; for all n > max

Ar1� Ar

;B

;

where the constants in the equivalence depend only on f and r.(ii) In the hypothesis of Theorem 2, if r < r1 < r1 þ 1 < 1=A and if f is not a polynomial of degree 6 p; ðp P 1Þ then

k½Sða;bÞn �ðpÞðf Þ � f ðpÞkr �

1n; for all n > max

Ar1� Ar

;B

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

Proof.

(i) For all jzj 6 r and n 2 N, we can write

Sða;bÞn ðf ÞðzÞ � f ðzÞ ¼ 1nða� bzÞf 0ðzÞ þ zf 00ðzÞ þ 1

n� n2 Sða;bÞn ðf ÞðzÞ � f ðzÞ � a� bz

nf 0ðzÞ � z

nf 00ðzÞ

� �� �

Applying the inequality

kF þ GkP j kFk � kGk jP kFk � kGk;

we obtain

kSða;bÞn ðf Þ � fkr P1n

�ða� be1Þf 0 þ e1f 00k kr�

1n� n2 Sða;bÞn ðf Þ � f � ða� be1Þf 0 þ e1f 00

n

r

�:

Since f is not a polynomial of degree 6 0 (i.e. a constant function) in DR, we get ða� be1Þf 0 þ e1f 00k kr > 0. Indeed, supposingthe contrary, it follows that

ða� bzÞf 0ðzÞ þ zf 00ðzÞ ¼ 0; for all jzj 6 r:

Denoting f 0ðzÞ ¼ yðzÞ, we get ða� bzÞyðzÞ þ zy0ðzÞ ¼ 0; for all jzj 6 r. Since yðzÞ is analytic, let yðzÞ ¼P1

k¼0bkzk. Replacing inthe above equation, by the coefficients identification we easily arrive at system ðaþ 1Þb0 ¼ 0; ðaþ 1þ kÞbk ¼ bbk�1;

k ¼ 1;2; . . . ;.Therefore we easily get bk ¼ 0 for all k ¼ 0;1; . . ., and yðzÞ ¼ 0, for all jzj 6 r. This implies that f is a constant function, in con-tradiction with the hypothesis.Now by Theorem 2, there exists a constant C > 0 independent of n, such that we have

314 S.G. Gal, V. Gupta / Applied Mathematics and Computation 234 (2014) 309–315

n2 Sða;bÞn ðf Þ � f � ða� be1Þf 0 þ e1f 00

n

r

6 C; for all n > maxAr

1� Ar;B

:

Thus, there exists n0 > max Ar1�Ar ;B� �

such that for all n P n0, we have

ða� be1f 0 þ e1f 00k kr �1n� n2 Sða;bÞn ðf Þ � f � ða� be1Þf 0 þ e1f 00

n

r

� �P

12ða� be1Þf 0 þ e1f 00k kr ;

which implies that

kSða;bÞn ðf Þ � fkr P1

2nða� be1Þf 0 þ e1f 00k kr

for all n P n0.For max Ar

1�Ar ;B� �

< n 6 n0 � 1, we get kSða;bÞn ðf Þ � fkr P Mr;nðf Þn with Mr;nðf Þ ¼ n � kSða;bÞn ðf Þ � fkr > 0 (since kSða;bÞn ðf Þ � fkr ¼ 0 for

a certain n is valid only for f a constant function, contradicting the hypothesis on f).Therefore, finally we have

jjSða;bÞn ðf Þ � f jjr PCrðf Þ

n;

for all n > Ar1�Ar ;B� �

, where

Crðf Þ ¼ minn0�1Pn>maxfAr=ð1�ArÞ;Bg

Mr;nðf Þ; . . . :;Mr;n0�1ðf Þ;12ða� be1Þf 0 þ e1f 00k kr

;

which combined with Theorem 1, (i), proves the desired conclusion.(ii) The upper estimate is exactly Theorem 1, (ii), therefore it remains to prove the lower estimate. Denote by C the circle

of radius r1 and center 0. By the Cauchy’s formulas for all jzj 6 r and n 2 N we get

½Sða;bÞn �ðpÞðf ÞðzÞ � f ðpÞðzÞ ¼ p!

2pi

ZC

Sða;bÞn ðf ÞðvÞ � f ðvÞðv � zÞpþ1 dv ;

where jv � zjP r1 � r for all jzj 6 r and v 2 C.For all v 2 C and n 2 N we get

Sða;bÞn ðf ÞðvÞ � f ðvÞ ¼ 1nða� bvÞf 0ðvÞ þ vf 00ðvÞ þ 1

nn2 Sða;bÞn ðf ÞðvÞ � f ðvÞ � ða� bvÞf 0ðvÞ þ vf 00ðvÞ

n

� �� � ;

which replaced in the Cauchy’s formula implies

½Sða;bÞn �ðpÞðf ÞðzÞ � f ðpÞðzÞ ¼ 1

np!

2pi

ZC

ða� bvÞf 0ðvÞ þ vf 00ðvÞðv � zÞpþ1 dv þ 1

n� p!

2pi

ZC

n2 Sða;bÞn ðf ÞðvÞ � f ðvÞ � ða�bvÞf 0ðvÞþvf 00ðvÞn

� �ðv � zÞpþ1 dv

8<:

9=;

¼ 1n

ða� bzÞf 0ðzÞ þ zf 00ðzÞ� �ðpÞ þ 1

n� p!

2pi

ZC

n2 Sða;bÞn ðf ÞðvÞ � f ðvÞ � ða�bvÞf 0ðvÞþvf 00ðvÞn

� �ðv � zÞpþ1 dv

8<:

9=;:

Passing to the norm k � kr , for all n 2 N we obtain

k½Sða;bÞn �ðpÞðf Þ � f ðpÞkr P

1n

ða� be1Þf 0 þ e1f 00½ �ðpÞ

r� 1

np!

2p

ZC

n2 Sða;bÞn ðf ÞðvÞ � f ðvÞ � ða�bvÞf 0ðvÞþvf 00ðvÞn

� �ðv � zÞpþ1 dv

r

8<:

9=;;

where by Theorem 2, for all n > max Ar1�Ar ;B� �

it follows

p!

2p

ZC

n2 Sða;bÞn ðf ÞðvÞ � f ðvÞ � ða�bvÞf 0 ðvÞþvf 00 ðvÞn

� �ðv � zÞpþ1 dv

r

6p!

2p� 2pr1n2

ðr1 � rÞpþ1 Snðf Þ � f � ða� be1Þf 0 þ e1f 00

n

r1

6 C � p!r1

ðr1 � rÞpþ1 :

Now, by hypothesis on f we have ða� be1Þf 0 þ e1f 00½ �ðpÞ

r> 0. Indeed, supposing the contrary it follows that

ða� bzÞf 0ðzÞ þ zf 00ðzÞ ¼ Q p�1ðzÞ is a polynomial of degree 6 p� 1. Denoting again yðzÞ ¼ f 0ðzÞ we arrive at the linear equationða� bzÞyðzÞ þ zy0ðzÞ ¼ Qp�1ðzÞ, with its the homogenous equation having only the zero solution.

S.G. Gal, V. Gupta / Applied Mathematics and Computation 234 (2014) 309–315 315

In consequence, clearly that this equation in real variable x can have as solution yðxÞ only polynomials of degree p� 1 in x,and from the analyticity of y and the identity theorem, the equation in the z variable will have as solution only polynomial ofdegree p� 1 in z. This implies that f necessarily is a polynomial of degree 6 p, in contradiction with the hypothesis.

For the rest of the proof, reasoning exactly as in the proof of the above point (i), we immediately get the requiredconclusion. h

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