Fractional calculus and some propertiesof certain starlike functions with

negative coefficients

Halit Orhan *, Muhammet Kamali

Atat€uurk €UUniversitesi Fen-Edebiyat Fak€uultesi, Matematik B€ool€uum€uu, 25240 Erzurum, Turkey

Abstract

A certain subclass Tcðn; p; k; aÞ of starlike functions in the unit disk is introduced. Themain object of this paper is to derive several interesting properties of functions be-

longing to the class Tcðn; p; k; aÞ. Various distortion inequalities for fractional calculus offunctions in the Tcðn; p; k; aÞ are also given.� 2002 Elsevier Science Inc. All rights reserved.

Keywords: Fractional calculus; Starlike functions; Hadamard product; Fractional integrals and

derivatives; Analytic functions

1. Introduction

Let T ðn; pÞ denote the class of functions f ðzÞ of the form:

f ðzÞ ¼ zp �X1k¼n

akþpzkþp akþp

�P 0; p 2 N :¼ f1; 2; 3; . . .g; n 2 N

�;

ð1:1Þ

which are analytic in the open unit disk

U ¼ z : z 2 C and jzjf < 1g:

* Corresponding author.

E-mail address: horhan@atauni.edu.tr (H. Orhan).

0096-3003/02/$ - see front matter � 2002 Elsevier Science Inc. All rights reserved.

PII: S0096 -3003 (02 )00037-1

Applied Mathematics and Computation 136 (2003) 269–279

www.elsevier.com/locate/amc

A function f ðzÞ 2 T ðn; pÞ is said to be in the class Tcðn; p; k; aÞ if it satisfiesthe inequality:

Rekcz3f 000ðzÞ þ ð2kc þ k � cÞf 00ðzÞ þ zf 0ðzÞ

kcz2f 00ðzÞ þ ðk � cÞzf 0ðzÞ þ ð1� k þ cÞf ðzÞ

� �> a ð1:2Þ

for some a ð06 a < 1Þ and k ð06 c6 k6 1Þ, and for all z 2 U . We note that

T0ðn; 1; 0; aÞ � TaðnÞ; ð1:3ÞT0ðn; 1; 1; aÞ � CaðnÞ; ð1:4ÞT0ð1; 1; 0; aÞ � T ðaÞ; ð1:5ÞT0ð1; 1; 1; aÞ � CðaÞ; ð1:6ÞT0ðn; 1; k; aÞ � Pðn; k; aÞ; ð1:7Þ

and

T0ðn; p; k; aÞ � T ðn; p; k; aÞ: ð1:8Þ

The classes TaðnÞ and CaðnÞ were studied earlier by Srivastava et al. [1], theclasses

T ðaÞ ¼ Tað1Þ and CðaÞ ¼ Cað1Þ ð1:9Þ

were studied by Silverman [2], and the class P ðn; k; aÞ was studied by Altıntas�[3], the class T ðn; p; k; aÞ was studied by Altıntas� et al. [7]. The class Tcðn; 1; k; aÞwas studied by Kamali and Kadıo�gglu [8].The object of the present paper is to give various basic properties of func-

tions belonging to the general class Tcðn; p; k; aÞ. We also prove (in Section 3)several distortion theorems (involving certain operators of fractional calculus)

for functions in the class Tcðn; p; k; aÞ.

2. A theorem on coefficient bounds

We begin by proving some sharp coefficient inequalities contained in the

following theorem.

Theorem 1. A function f ðzÞ 2 T ðn; pÞ is in the class Tcðn; p; k; aÞ if and only if

X1k¼n

ðk þ p � aÞ ðk½ þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1�akþp

6 ðp � aÞ ðp½ � 1Þðkcp þ k � cÞ þ 1�; ð06 a < 1; 06 c6 k6 1

ðp � aÞðp � 1Þðkcp þ k � cÞP aðp 6¼ 1Þ; p; n 2 NÞ:

ð2:1Þ

The result is sharp.

270 H. Orhan, M. Kamali / Appl. Math. Comput. 136 (2003) 269–279

Proof. Suppose that f ðzÞ 2 Tcðn; p; k; aÞ. Then we find from (1.2) that

Rep ðp � 1Þðkcp þ k � cÞ þ 1½ �zp �

P1k¼nðk þ pÞ ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ �akþpzkþp

ðp � 1Þ ðkcp þ k � cÞ þ 1½ �zp �P1

k¼n ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ �akþpzkþp

� �> a

ð06 a < 1; 06 c6 k6 1; ðp � aÞðp � 1Þðkcp þ k � cÞP aðp 6¼ 1Þ; p; n 2 N; z 2 UÞ:

If we choose z to be real and let z ! 1�, we get

Reðp � 1Þ½ðkcp þ k � cÞ þ 1� �

P1k¼nðk þ pÞ ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ �akþp

ðp � 1Þ ðkcp þ k � cÞ þ 1½ � �P1

k¼n ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ �akþp

� �P a

ð06 a < 1; 06 c6 k6 1; ðp � aÞðp � 1Þðkcp þ k � cÞP aðp 6¼ 1Þ; p; n 2 NÞ

or, equivalently,

X1k¼n

ðk þ p � aÞ ðk½ þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1�akþpðp � aÞ

� ðp½ � 1Þðkcp þ k � cÞ þ 1�ð06 a < 1; 06 c6 k6 1; ðp � aÞðp � 1Þðkcp þ k � cÞP aðp 6¼ 1Þ; p; n 2 NÞ;

which is precisely the assertion (2.1) of Theorem 1.Conversely, suppose that the inequality (2.1) holds true and let

z 2 oU ¼ z : z 2 C and jzjf ¼ 1g:Then we find from the definition (1.1) that

kcz3f 000ðzÞ þ ð2kc þ k � cÞz2f 00ðzÞ þ zf 0ðzÞkcz2f 00ðzÞ þ ðk � cÞzf 0ðzÞ þ ð1� k þ cÞf ðzÞ

���� � ðp � aÞ ðp½ � 1Þðkcp þ k � cÞ þ 1�����

¼����� ðp½ � 1Þðkcp þ k � cÞ þ 1� ðp½ � aÞðp � 1Þðkcp þ k � cÞ � a�zp

�X1k¼n

ðk½ þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1� ðk½ þ aÞ

� ðp � aÞðp � 1Þðkcp þ k � cÞ�akþpzkþp

�����

ðp½���� � 1Þðkcp þ k � cÞ þ 1�zp

�X1k¼n

ðk½ þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1�akþpzkþp

����6 ðp½(

� 1Þðkcp þ k � cÞ þ 1� ðp½ � aÞðp � 1Þðkcp þ k � cÞ � a�jzjp

þX1k¼n

ðk½ þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1� ðk½ þ aÞ

� ðp � aÞðp � 1Þðkcp þ k � cÞ�akþpjzjkþp

),ðp½ � 1Þðkcp þ k � cÞ þ 1�jzjp

�X1k¼n

½ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1� akþpjzjkþp

6 ðp � aÞ ðp½ � 1Þðkcp þ k � cÞ� � a ð06 a < 1; 06 c6 k6 1;ðp � aÞðp � 1Þðkcp þ k � cÞP aðp 6¼ 1Þ; z 2 oU ; p; n 2 NÞ;

H. Orhan, M. Kamali / Appl. Math. Comput. 136 (2003) 269–279 271

provided that the inequality (2.1) is satisfied. Hence, by the maximum modulus

theorem, we have

f ðzÞ 2 Tcðn; p; k; aÞ:Finally, we note the assertion (2.1) of Theorem 1 is sharp, the extremal func-

tion being

f ðzÞ ¼ zp � ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ � z

nþp

ðn; p 2 NÞ: �

ð2:2Þ

Theorem 2. Let the function f ðzÞ defined by (1.1) and the function gðzÞ definedby

gðzÞ ¼ zp �X1k¼n

bkþpzkþp ðbkþp P 0; p; n 2 NÞ ð2:3Þ

be in the same class Tcðn; p; k; aÞ. Then the function hðzÞ defined by

hðzÞ ¼ ð1� bÞf ðzÞ þ bgðzÞ ¼ zp �X1k¼n

ckþpzkþp;

ckþp :¼ ð1� bÞakþp þ bbkþp P 0; 06 b6 1; p 2 N

ð2:4Þ

is also in the class Tcðn; p; k; aÞ.

Proof. Suppose that each of the functions f ðzÞ and gðzÞ is in the class

Tcðn; p; k; aÞ. Then, making use of (2.1), we see thatX1k¼n

ðkþp�aÞ ðk½ þ p�1Þðkcðkþ pÞþk� cÞþ1�ckþp

¼ð1�bÞX1k¼n

ðkþp�aÞ ðk½ þ p�1Þðkcðkþ pÞþk� cÞþ1�akþp

þbX1k¼n

ðkþp�aÞ ðk½ þ p�1Þðkcðkþ pÞþk� cÞþ1�bkþp

6ð1�bÞðp�aÞ ðp½ �1Þðkcpþk� cÞþ1�þbðp�aÞ ðp½ �1Þðkcpþk�cÞþ1�¼ ðp�aÞ ðp½ �1Þðkcpþk� cÞþ1�ð06a< 1; 06c6k61; ðp�aÞðp�1Þðkcpþk� cÞPaðp 6¼ 1Þ; p;n2NÞ;

ð2:5Þ

which completes the proof of Theorem 2. �

Next we define the modified Hadamard product of functions f ðzÞ and gðzÞ,which are defined by (1.1) and (2.3), respectively, by

272 H. Orhan, M. Kamali / Appl. Math. Comput. 136 (2003) 269–279

f gðzÞ ¼ zp �X1k¼n

akþpbkþpzkþp ð2:6Þ

ðakþp P 0; bkþp P 0; p 2 NÞ:

Theorem 3. If each of the functions f ðzÞ and gðzÞ is in the class Tcðn; p; k; aÞ, then

f gðzÞ 2 Tcðn; p; k; dÞ;

where

d6 p � ðp � aÞ2 ðp � 1Þðkcp þ k � cÞ þ 1½ �ðk þ p � aÞ ðk þ p � 1Þðkcðp þ 1Þ þ k � cÞ þ 1½ �

ðn; p 2 NÞ:ð2:7Þ

The result is sharp for the functions f ðzÞ and gðzÞ given by

f ðzÞ ¼ gðzÞ

¼ zp � ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �

ðn; p 2 NÞ:

ð2:8Þ

Proof. From Theorem 1, we have

X1k¼n

ðk þ p � aÞ ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ �ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ � akþp 6 1

ðn; p 2 NÞð2:9Þ

and

X1k¼n

ðk þ p � aÞ ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ �ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ � bkþp 6 1

ðn; p 2 NÞ:ð2:10Þ

We have to find the largest d such that

X1k¼n

ðk þ p � dÞ ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ �ðp � dÞ ðp � 1Þðkcp þ k � cÞ þ 1½ � akþpbkþp 6 1

ðn; p 2 NÞ: ð2:11Þ

From (2.9) and (2.10) we find, by means of the Cauchy–Schwarz inequality,

that

H. Orhan, M. Kamali / Appl. Math. Comput. 136 (2003) 269–279 273

X1k¼n

ðk þ p � aÞ ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ �ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiakþpbkþp

p6 1

ðp 2 NÞ: ð2:12Þ

Therefore, (2.11) holds true if

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiakþpbkþp

p6

p � dp � a

ðkP n; n; p 2 NÞ; ð2:13Þ

that is, if

X1k¼n

ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðk þ p � aÞ ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ � 6

p � dp � a

ðk P n; n; p 2 NÞ;ð2:14Þ

which readily yields

d6 p � ðp � aÞ2 ðp � 1Þðkcp þ k � cÞ þ 1½ �ðk þ p � aÞ ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ �

ðk P n; n; p 2 NÞ:ð2:15Þ

Finally, letting

/ðkÞ ¼ p � ðp � aÞ2 ðp � 1Þðkcp þ k � cÞ þ 1½ �ðk þ p � aÞ ðk þ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ �

ðk P n; n; p 2 NÞ;ð2:16Þ

we see that the function /ðkÞ is increasing in k. This shows that

d6/ðnÞ ¼ p � ðp � aÞ2 ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �

ðp 2 NÞ; ð2:17Þ

which completes the proof of Theorem 3. �

Corollary 1. If f ðzÞ 2 Tcðn; p; k; cÞ, then

anþp 6ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �

ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �ðn; p 2 NÞ:

ð2:18Þ

Numerous consequences of Theorems 1–3 (and of Corollary 1) can indeed

be deduced by specializing the various parameters involved. Many of these

consequences were proven by earlier workers on the subject (cf. eg. [1–3]).

274 H. Orhan, M. Kamali / Appl. Math. Comput. 136 (2003) 269–279

3. Distortion theorems involving operators of fractional calculus

In this section, we shall prove several distortion theorems for functions

belonging to the general class Tcðn; p; k; aÞ. Each of these theorems would in-volve certain operators of fractional calculus, which are defined as follows (cf.

eg. [4–6,9]).

Definition 1. The fractional integral of order l is defined by

D�lz f ðzÞ ¼

Z z

0

f ðfÞðz� fÞ1�l df ðl > 0Þ;

where f ðzÞ is an analytic function in a simply connected region of the z-planecontaining the origin, and the multiplicity of ðz� fÞl�1 is removed by requiringlogðz� fÞ to be real when z� f > 0.

Definition 2. The fractional derivative of order l is defined by

Dlz f ðzÞ ¼

1

Cð1� lÞd

dz

Z z

0

f ðfÞðz� fÞl df ð06 l < 1Þ;

where f ðzÞ is constrained, and multiplicity of ðz� fÞ�lis removed, as in Def-

inition 1.

Definition 3. Under the hypotheses of Definition 1, the fractional derivative oforder nþ l is defined by

Dnþlz f ðzÞ ¼ dn

dznDl

z f ðzÞ ð06 l < 1; n 2 N0 :¼ N [ f0gÞ:

Theorem 4. If f ðzÞ 2 Tcðn; p; k; aÞ, then

D�lz f ðzÞ

�� ��6 jzjpþl Cðp þ 1Þ

Cðp þ l þ 1Þ

�

þ ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �Cðnþ p þ l þ 1Þð Þ jzj

�;

ð3:1Þ

H. Orhan, M. Kamali / Appl. Math. Comput. 136 (2003) 269–279 275

and

D�lz f ðzÞ

�� ��P jzjpþl Cðp þ 1Þ

Cðp þ l þ 1Þ

�

� ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �Cðnþ p þ l þ 1Þð Þ jzj

�;

ð3:2Þ

for l > 0; n 2 N, and p 2 N, and for all z 2 U .The result is sharp for the function f ðzÞ given by

f ðzÞ ¼ zp � ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðk þ pÞ þ k � cÞ þ 1½ � z

nþp

ðn; p 2 NÞ:ð3:3Þ

Proof. Suppose that f ðzÞ 2 Tcðn; p; k; aÞ. We then find from (2.1) that

X1k¼n

akþp 6ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �

ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �ðn; p 2 NÞ: ð3:4Þ

Making use of (3.4) Definition 1, we have

D�lz f ðzÞ ¼ zpþl Cðp þ 1Þ

Cðp þ l þ 1Þ

(�X1k¼n

Cðk þ l þ 1ÞCðk þ p þ l þ 1Þ akþpzk

)

¼ zpþl Cðp þ 1ÞCðp þ l þ 1Þ

(�X1k¼n

wðkÞakþpzk); ð3:5Þ

where, for convenience,

wðkÞ ¼ Cðk þ p þ 1ÞCðk þ p þ l þ 1Þ ðl > 0; kP n; n; p 2 NÞ:

Clearly, the function wðkÞ is decreasing in k, and we have

0 < wðkÞ6wðnÞ ¼ Cðk þ p þ 1ÞCðk þ p þ l þ 1Þ : ð3:6Þ

276 H. Orhan, M. Kamali / Appl. Math. Comput. 136 (2003) 269–279

Thus we find from (3.4)–(3.6) that

D�lz f ðzÞ

�� ��6 jzjpþl Cðp þ 1Þ

Cðp þ l þ 1Þ

(þ jzjwðnÞ

X1k¼n

akþp

)6 jzjpþl Cðp þ 1Þ

Cðp þ l þ 1Þ

�

þ ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �Cðnþ p þ l þ 1Þð Þ jzj

�;

which is precisely the assertion (3.1), and that

D�lz f ðzÞ

�� ��P jzjpþl Cðp þ 1Þ

Cðp þ l þ 1Þ

(� jzjwðnÞ

X1k¼n

akþp

)P jzjpþl Cðp þ 1Þ

Cðp þ l þ 1Þ

�

� ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �Cðnþ p þ l þ 1Þð Þ jzj

�;

which is the same as assertion (3.2). �

In order to complete the proof of Theorem 4, it is easily observed that the

equalities in (3.1) and (3.2) are satisfied by the function f ðzÞ given by (3.3).The proofs of Theorems 5 and 6 below are much akin to that of Theorem 4,

which we have detailed above fairly fully. Indeed, instead of Definition 1, we

make use of Definition 2 and 3 to prove Theorems 5 and 6, respectively.

Theorem 5. If f ðzÞ 2 Tcðn; p; k; aÞ, thenDl

z f ðzÞ�� ��

P jzjp�l Cðp þ 1ÞCðp � l þ 1Þ

�

þ ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �Cðnþ p þ 1Þðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �Cðnþ p þ l þ 1Þð Þ jzj

�ð3:7Þ

and

Dlz f ðzÞ

�� ��P jzjp�l Cðp þ 1Þ

Cðp � l þ 1Þ

�

� ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �Cðnþ p þ 1Þðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �Cðnþ p þ l þ 1Þð Þ jzj

�ð3:8Þ

for 06 l < 1, n 2 N, and p 2 N, and for all z 2 U .The result is sharp for the function f ðzÞ given by (3.3).

H. Orhan, M. Kamali / Appl. Math. Comput. 136 (2003) 269–279 277

Theorem 6. If f ðzÞ 2 Tcðn; p; k; aÞ, then

D1þlz f ðzÞ

�� ��6 jzjp�l�1 Cðp þ 1Þ

Cðp � lÞ

�

þ ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �Cðnþ p þ 1Þðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �Cðnþ p � lÞð Þ jzj

�ð3:9Þ

and

D1þlz f ðzÞ

�� ��P jzjp�l�1 Cðp þ 1Þ

Cðp � lÞ

�

� ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �Cðnþ p þ 1Þðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ �Cðnþ p � lÞð Þ jzj

�ð3:10Þ

for 06 l < 1, n 2 N and p 2 N, and for all z 2 U .The result is sharp for the function f ðzÞ given by (3.3).

Setting l ¼ 0 in Theorem 5, we obtain the following corollary.

Corollary 2. If f ðzÞ 2 Tcðn; p; k; aÞ, then

f ðzÞj j6 jzjp 1

�þ ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ � jzj

�ð3:11Þ

and

f ðzÞj jP jzjp 1

�� ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ � jzj

�ð3:12Þ

for n 2 N and p 2 N, and for all z 2 U .The result is sharp for the function f ðzÞ given by (3.3).

If, on the other hand, we set l ¼ 0 in Theorem 6, we shall arrive at

Corollary 3.

278 H. Orhan, M. Kamali / Appl. Math. Comput. 136 (2003) 269–279

Corollary 3. If f ðzÞ 2 Tcðn; p; k; aÞ, then

f 0ðzÞ�� ��6 jzjp�1 p

�þ ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ � jzj

�ð3:13Þ

and

f 0ðzÞ�� ��P jzjp�1 p

�� ðp � aÞ ðp � 1Þðkcp þ k � cÞ þ 1½ �ðnþ p � aÞ ðnþ p � 1Þðkcðnþ pÞ þ k � cÞ þ 1½ � jzj

�ð3:14Þ

for n 2 N and p 2 N, and for all z 2 U .The result is sharp for the function f ðzÞ given by (3.3).

Further consequences of distortion properties (given by Corollaries 2 and 3)

can be obtained for each of the function classes studied by earlier workers. The

details may be omitted.

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