Neighborhood properties of certain classes of multivalently analytic functions associated with the convolution structure

Applied Mathematics and Computation 218 (2012) 6511–6518

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Neighborhood properties of certain classes of multivalently analytic functions associated with the convolution structure H.M. Srivastava a,⇑, Serap Bulut b a b

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada _ Department of Mathematics, Civil Aviation College, Kocaeli University (Arslanbey Campus), TR-41285 Izmit-Kocaeli, Turkey

a r t i c l e

i n f o

a b s t r a c t

Keywords: Analytic functions Starlike and convex functions Multivalent functions Hadamard product (or convolution) Coefficient bounds Distortion inequalities Neighborhood properties Non-homogeneous Cauchy–Euler differential equations

In this paper, by making use of the familiar concept of neighborhoods of p-valently analytic functions, we prove coefficient bounds, distortion inequalities and associated inclusion relations for the (n, d)-neighborhoods of a family of p-valently analytic functions and their derivatives, which is defined by means of a certain general family of non-homogenous Cauchy–Euler differential equations. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction and definitions Let R ¼ ð1; 1Þ be the set of real numbers, C be the set of complex numbers,

N :¼ f1; 2; 3; . . .g ¼ N0 n f0g be the set of positive integers and

N :¼ N n f1g ¼ f2; 3; 4; . . .g: Let T n ðpÞ denote the class of functions of the form:

f ðzÞ ¼ zp 

1 X

ak zk

ðak = 0; p; n 2 N :¼ f1; 2; 3; . . .gÞ;

ð1:1Þ

k¼nþp

which are analytic and p-valent in the open unit disk

U ¼ fz : z 2 C and jzj < 1g: A function f 2 T n ðpÞ is said to be p-valently starlike of order a (0 5 a < p), that is, f 2 T n;p ðaÞ if it satisfies the following inequality:

R

 0  zf ðzÞ > a ðz 2 UÞ: f ðzÞ

Furthermore, a function f 2 T n ðpÞ is said to be p-valently convex of order a (0 5 a < p), that is, f 2 C n;p ðaÞ if it satisfies the following inequality: ⇑ Corresponding author. E-mail addresses: [email protected] (H.M. Srivastava), [email protected] (S. Bulut). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.12.022

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H.M. Srivastava, S. Bulut / Applied Mathematics and Computation 218 (2012) 6511–6518

  00 zf ðzÞ > a ðz 2 UÞ: R 1þ 0 f ðzÞ Denote by f ⁄ g the Hadamard product (or convolution) of the functions f and g, that is, if f is given by (1.1) and g is given by 1 X

gðzÞ ¼ zp þ

b k zk

ðp; n 2 NÞ;

ð1:2Þ

k¼nþp

then

ðf  gÞðzÞ :¼ zp 

1 X

ak bk zk ¼: ðg  f ÞðzÞ ðp; n 2 NÞ:

ð1:3Þ

k¼nþp

Let S n;p ðg; k; l; aÞ denote the subclass of T n ðpÞ consisting of functions f which satisfy the following inequality:

R

  zðF k;l  gÞ0 ðzÞ > a ð0 5 l 5 k 5 1; 0 5 a < p; z 2 UÞ; ðF k;l  gÞðzÞ

ð1:4Þ

where 0

F k;l ðzÞ ¼ klz2 f 00 ðzÞ þ ðk  lÞzf ðzÞ þ ð1  k þ lÞf ðzÞ

ð1:5Þ

and g is given by (1.2). Various special cases of the class S n;p ðg; k; l; aÞ were considered by many earlier researchers on this topic of Geometric Function Theory. For example, we have the following relationships with the classes which were studied in some of these earlier works:

S n;p ðg; k; 0; aÞ ¼ S n;p ðg; k; aÞ :¼

    zðF k  g Þ0 ðzÞ > a ð0 5 a < p; z 2 U; F k ¼ F k;0 Þ ; f : f 2 T n ðpÞ and R ðF k  g ÞðzÞ

 z  ; k; l; a  T l ðn; p; k; aÞ ðsee ½7Þ; 1z  z  S n;p ; 0; 0; a  T n;p ðaÞ  T a ðp; nÞ ðsee ½13Þ; 1z !  z  z  S n;p ; 1; 0; a  C n;p ðaÞ  CT a ðp; nÞ ðsee ½13Þ; ; 0; 0; a S n;p 2 1z ð1  zÞ  z  ; k; 0; a  T n ðp; a; kÞ ðsee ½3Þ; S n;p 1z  z  ; k; 0; a  Pðn; k; aÞ ðsee ½2Þ; S n;1 1z  z  S 1;p ; 0; 0; a  T  ðp; aÞ ðsee ½8Þ; 1z !  z  z ; 1; 0; ; 0; 0; a a  Cðp; aÞ ðsee ½8Þ; S 1;p  S 1;p 1z ð1  zÞ2  z  ; 0; 0; a  T a ðnÞ ðsee ½12Þ; S n;1 1z !  z  z  S n;1 ; 1; 0; a  C a ðnÞ ðsee ½12Þ; ; 0; 0; a S n;1 2 1z ð1  zÞ  z  S 1;1 ; 0; 0; a  T  ðaÞ ðsee ½10Þ 1z S n;p

and

!

z

S 1;1

; 0; 0; a ð1  zÞ2

 S 1;1

ð1:6Þ ð1:7Þ ð1:8Þ

 z  ; 1; 0; a  CðaÞ ðsee ½10Þ: 1z

Finally, let Rn;p ðg; k; l; a; m; uÞ denote the subclass of T n ðpÞ consisting of functions f which satisfy the following non-homogenous Cauchy–Euler differential equation (see, for example, [11, p. 1360, Eq. (9)]): m

zm

d w m þ dz

¼ hðzÞ



   m1 m1 Y d w m m ðu þ m  1Þzm1 m1 þ    þ w ðu þ jÞ 1 m dz j¼0

m1 Y

ðu þ j þ pÞ

j¼0



w ¼ f ðzÞ 2 T n ðpÞ; h 2 S n;p ðg; k; l; aÞ; m 2 N and u 2 ðp; 1Þ :

ð1:9Þ

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H.M. Srivastava, S. Bulut / Applied Mathematics and Computation 218 (2012) 6511–6518

For

gðzÞ ¼

z ; 1z

m ¼ 2 and

l ¼ 0;

we have the class given by



Rn;p ðg; k; l; a; uÞ  Kn ðp; a; k; uÞ

w ¼ f ðzÞ 2 T n ðpÞ; h 2 T n ðp; a; kÞ and u 2 ðp; 1Þ ;

ð1:10Þ

which was introduced and studied by Altıntasß et al. [5]. Following the works of Goodman [6] and Ruscheweyh [9] (see also [4,5]), Altıntasß [1] defined the (n, d)-neighborhood of a function f 2 T n ðpÞ by

(

d N n;p ðf Þ

h : h 2 T n ðpÞ;

¼

p

hðzÞ ¼ z 

1 X

k

ck z and

k¼nþp

1 X

)

kjak  ck j 5 d :

ð1:11Þ

k¼nþp

It follows from the definition (1.11) that, if

eðzÞ ¼ zp

ðp 2 NÞ;

ð1:12Þ

then

( d N n;p ðeÞ

¼

h : h 2 T n ðpÞ;

p

hðzÞ ¼ z 

1 X

k

ck z and

k¼nþp

1 X

) kjck j5d :

ð1:13Þ

k¼nþp

The main object of this paper is to investigate the various properties and characteristics of functions belonging to the above-defined classes

S n;p ðg; k; l; aÞ and Rn;p ðg; k; l; a; m; uÞ: Apart from deriving coefficient bounds and distortion inequalities for each of these function classes, we establish several inclusion relationships involving the (n, d)-neighborhoods of functions belonging to the general function classes which are introduced above. 2. Coefficient bounds and distortion inequalities In this section, we prove the following results which yield the distortion inequalities for functions in the subclasses

S n;p ðg; k; l; aÞ and Rn;p ðg; k; l; a; m; uÞ: Lemma 1. Let the function f 2 T n ðpÞ be defined by (1.1). Then f is in the class S n;p ðg; k; l; aÞ if and only if 1 X

ðk  aÞwðkÞak bk 5 ðp  aÞwðpÞ ð0 5 l 5 k 5 1; 0 5 a < p; z 2 UÞ;

ð2:1Þ

k¼nþp

where

wðsÞ ¼ ðs  1Þðkls þ k  lÞ þ 1:

ð2:2Þ

The result is sharp for the function f given by

f ðzÞ ¼ zp 

ðp  aÞwðpÞ k z ðk  aÞwðkÞ

ðk ¼ n þ p; n þ p þ 1; . . .Þ:

ð2:3Þ

Proof. We first suppose that the function f given by (1.1) is in the class S n;p ðg; k; l; aÞ. Then, in conjunction with (1.4) and (1.5), we find that

R

pwðpÞzp  wðpÞzp 

P1

k¼nþp kwðkÞak bk z P1 k k¼nþp wðkÞak bk z

k

!

> a:

If we choose z to be real and let z ? 1, we arrive easily at the inequality (2.1). Conversely, we suppose that the inequality (2.1) holds true and let

z 2 @U ¼ fz : z 2 C and jzj ¼ 1g: We then find from the Eqs. (1.3), (1.4) and (1.5) that

P1

P1 P



k k



zðF k;l  gÞ0 ðzÞ

pwðpÞzp  1



k¼nþp kwðkÞak bk z k¼nþp ðk  pÞwðkÞak bk z

k¼nþp ðk  pÞwðkÞak bk



ðF  gÞðzÞ  p ¼

wðpÞzp  P1 wðkÞa b zk  p

¼

wðpÞzp  P1 wðkÞa b zk

5 wðpÞ  P1 wðkÞa b : k;l k k k k k k k¼nþp k¼nþp k¼nþp

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H.M. Srivastava, S. Bulut / Applied Mathematics and Computation 218 (2012) 6511–6518

By means of the inequality (2.1), we can write 1 X

ðk  pÞwðkÞak bk 5 ðp  aÞwðpÞ 

k¼nþp

1 X

ðp  aÞwðkÞak bk :

k¼nþp

We thus obtain

  P1

 P1

z F  g 0 ðzÞ ðp  a Þ wðpÞ  wðkÞa b k k ðk  pÞwðkÞa b k¼nþp k k

 k;l k¼nþp P P 5 ¼ p  a:  p 5



F k;l  g ðzÞ wðpÞ  1 wðpÞ  1 k¼nþp wðkÞak bk k¼nþp wðkÞak bk This evidently completes the proof of Lemma 1. h Lemma 2. Let the function f given by (1.1) be in the class S n;p ðg; k; l; aÞ. Then 1 X

ak 5

k¼nþp

ðp  aÞwðpÞ ðn þ p  aÞwðn þ pÞbnþp

ðbk = bnþp Þ

ð2:4Þ

and 1 X

kak 5

k¼nþp

ðn þ pÞðp  aÞwðpÞ ðn þ p  aÞwðn þ pÞbnþp

ðbk = bnþp Þ:

ð2:5Þ

Proof. By using Lemma 1, we find from (2.1) that 1 X

ðn þ p  aÞwðn þ pÞbnþp

ak 5 ðp  aÞwðpÞ;

ð2:6Þ

k¼nþp

which immediately yields the first assertion (2.4) of Lemma 2. For the proof of second assertion of Lemma 2, by appealing to (2.1), we also obtain

wðn þ pÞbnþp

1 X

kak  a

k¼nþp

1 X

! ak

5 ðp  aÞwðpÞ:

ð2:7Þ

k¼nþp

Thus, by using (2.4) in (2.7), we can get the second assertion (2.5) of Lemma 2. h Our proposed distortion inequalities for functions in the multivalently analytic function class S n;p ðg; k; l; aÞ are given by Theorem 1 below. Theorem 1. Let a function f 2 T n ðpÞ be in the class S n;p ðg; k; l; aÞ. Then

jf ðzÞj 5 jzjp þ

ðp  aÞwðpÞ jzjnþp ðn þ p  aÞwðn þ pÞbnþp

ðz 2 UÞ

ð2:8Þ

jf ðzÞj = jzjp 

ðp  aÞwðpÞ jzjnþp ðn þ p  aÞwðn þ pÞbnþp

ðz 2 UÞ

ð2:9Þ

and

and (in general)

ðqÞ

f ðzÞ 5

p! ðn þ pÞ!ðp  aÞwðpÞ jzjnþpq jzjpq þ ðp  qÞ! ðn þ p  qÞ!ðn þ p  aÞwðn þ pÞbnþp

ðp > q; q 2 N0 ; z 2 UÞ

ð2:10Þ

ðqÞ

f ðzÞ =

p! ðn þ pÞ!ðp  aÞwðpÞ jzjpq  jzjnþpq ðp  qÞ! ðn þ p  qÞ!ðn þ p  aÞwðn þ pÞbnþp

ðp > q; q 2 N0 ; z 2 UÞ:

ð2:11Þ

and

The result is sharp for the function f given by (2.3). Proof. Suppose that f 2 S n;p ðg; k; l; aÞ. We then find from the inequality (2.4) that

jf ðzÞj 5 jzjp þ jzjnþp

1 X

ak 5 jzjp þ

k¼nþp

which is equivalent to (2.8), and that

ðp  aÞwðpÞ jzjnþp ðn þ p  aÞwðn þ pÞbnþp

ðz 2 UÞ;

H.M. Srivastava, S. Bulut / Applied Mathematics and Computation 218 (2012) 6511–6518

jf ðzÞj = jzjp  jzjnþp

1 X

ak = jzjp 

k¼nþp

ðp  aÞwðpÞ jzjnþp ðn þ p  aÞwðn þ pÞbnþp

6515

ðz 2 UÞ;

which is precisely the assertion (2.9). The demonstration of the inequalities (2.10) and (2.11), as well as of the assertion involving sharpness of the results of Theorem 1 is fairly straightforward. h The distortion inequalities for functions in the class Rn;p ðg; k; l; a; m; uÞ are given by Theorem 2 below. Theorem 2. Let a function f 2 T n ðpÞ be in the class Rn;p ðg; k; l; a; m; uÞ. Then

jf ðzÞj 5 jzjp þ

ðp  aÞwðpÞ

Qm1

ðu þ j þ pÞ jzjnþp Qm2 ðu þ j þ n þ pÞb nþp j¼0

j¼0

ðn þ p  aÞwðn þ pÞðm  1Þ

ðz 2 UÞ

ð2:12Þ

ðz 2 UÞ;

ð2:13Þ

and

jf ðzÞj = jzjp 

ðp  aÞwðpÞ

Qm1

ðu þ j þ pÞ jzjnþp Qm2 ðu þ j þ n þ pÞb nþp j¼0

j¼0

ðn þ p  aÞwðn þ pÞðm  1Þ

and (in general)

ðqÞ

f ðzÞ 5

p! jzjpq ðp  qÞ!

Q ðn þ pÞ!ðp  aÞwðpÞ m1 j¼0 ðu þ j þ pÞ þ jzjnþpq Q ðu þ j þ n þ pÞb ðn þ p  qÞ!ðn þ p  aÞwðn þ pÞðm  1Þ m2 nþp j¼0

ðp > q; q 2 N0 ; z 2 UÞ

ð2:14Þ

ðp > q; q 2 N0 ; z 2 UÞ:

ð2:15Þ

and

ðqÞ

f ðzÞ =

p! jzjpq ðp  qÞ!

Q ðn þ pÞ!ðp  aÞwðpÞ m1 j¼0 ðu þ j þ pÞ  jzjnþpq Q ðu þ j þ n þ pÞb ðn þ p  qÞ!ðn þ p  aÞwðn þ pÞðm  1Þ m2 nþp j¼0

Proof. Suppose that the function f 2 T n ðpÞ is given by (1.1). Also let the function h 2 S n;p ðg; k; l; aÞ be given as the function occurring in the general non-homogenous Cauchy–Euler differential equation (1.9) with, of course,

ck = 0 ðk ¼ n þ p; n þ p þ 1; n þ p þ 2; . . .Þ: We then readily see from (1.9) that

Qm1 j¼0

ðu þ j þ pÞ

j¼0

ðu þ j þ kÞ

ak ¼ Qm1

ck

ðk ¼ n þ p; n þ p þ 1; . . .Þ;

ð2:16Þ

so that

f ðzÞ ¼ zp 

1 X

ak zk ¼ zp 

k¼nþp

1 X k¼nþp

Qm1 j¼0

ðu þ j þ pÞ

j¼0

ðu þ j þ kÞ

Qm1

c k zk

ð2:17Þ

ðz 2 UÞ:

ð2:18Þ

and

jf ðzÞj 5 jzjp þ jzjnþp

1 X k¼nþp

Qm1 j¼0

ðu þ j þ pÞ

j¼0

ðu þ j þ kÞ

Qm1

ck

Moreover, since h 2 S n;p ðg; k; l; aÞ, the first assertion (2.4) of Lemma 2 yields the following inequality:

ck 5

ðp  aÞwðpÞ ðn þ p  aÞwðn þ pÞbnþp

ðk ¼ n þ p; n þ p þ 1; . . .Þ;

ð2:19Þ

which, together with (2.18) and (2.19), yields

jf ðzÞj 5 jzjp þ

Q 1 X ðp  aÞwðpÞ m1 1 j¼0 ðu þ j þ pÞ jzjnþp  Qm1 ðn þ p  aÞwðn þ pÞbnþp k¼nþp j¼0 ðu þ j þ kÞ

Finally, in view of the following sum:

ðz 2 UÞ:

ð2:20Þ

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H.M. Srivastava, S. Bulut / Applied Mathematics and Computation 218 (2012) 6511–6518 1 X k¼nþp

Qm1 j¼0

1 ðk þ u þ jÞ

¼ ¼

1 X

m1 X

k¼nþp

j¼0

ðm  1Þ

ð1Þj ðm  1  jÞ!j!ðk þ u þ jÞ

Qm2 j¼0

1 ðu þ j þ n þ pÞ

!

ðu 2 R n fn  p; n  p  1; n  p  2; . . .gÞ;

ð2:21Þ

the assertion (2.12) of Theorem 2 follows at once from (2.20) in conjunction with (2.21). The assertion (2.13) can be proven by similarly applying (2.17), (2.19) and (2.21). h Upon setting

gðzÞ ¼

z ; 1z

m ¼ 2 and

l¼0

in Theorem 2, if we use the relationship in (1.8), we obtain the following Corollary 1 given by Altıntasß et al. [5]. Corollary 1 [5, Theorem 1]. If the functions f and h satisfy the general non-homogeneous Cauchy–Euler differential equation. (1.9) with h 2 T n ðp; a; kÞ, then

jf ðzÞj 5 jzjp þ

ðp  aÞ½kðp  1Þ þ 1ðp þ uÞðp þ u þ 1Þ nþp jzj ðn þ p  aÞ½kðn þ p  1Þ þ 1ðn þ p þ uÞ

ðz 2 UÞ

ð2:22Þ

jf ðzÞj = jzjp 

ðp  aÞ½kðp  1Þ þ 1ðp þ uÞðp þ u þ 1Þ nþp jzj ðn þ p  aÞ½kðn þ p  1Þ þ 1ðn þ p þ uÞ

ðz 2 UÞ:

ð2:23Þ

and

By setting k = 0 and k = 1 in Corollary 1, and using the relationships in (1.6) and (1.7), we arrive at Corollaries 2 and 3, respectively. Corollary 2. If the functions f and h satisfy the general non-homogeneous Cauchy–Euler differential equation (1.9) with h 2 T n;p ðaÞ, then

jf ðzÞj 5 jzjp þ

ðp  aÞðp þ uÞðp þ u þ 1Þ nþp jzj ðn þ p  aÞðn þ p þ uÞ

ðz 2 UÞ

ð2:24Þ

jf ðzÞj = jzjp 

ðp  aÞðp þ uÞðp þ u þ 1Þ nþp jzj ðn þ p  aÞðn þ p þ uÞ

ðz 2 UÞ:

ð2:25Þ

and

Corollary 3. If the functions f and h satisfy the general non-homogeneous Cauchy–Euler differential equation (1.9) with h 2 C n;p ðaÞ, then

jf ðzÞj 5 jzjp þ

pðp  aÞðp þ uÞðp þ u þ 1Þ jzjnþp ðn þ pÞðn þ p  aÞðn þ p þ uÞ

ðz 2 UÞ

ð2:26Þ

jf ðzÞj = jzjp 

pðp  aÞðp þ uÞðp þ u þ 1Þ jzjnþp ðn þ pÞðn þ p  aÞðn þ p þ uÞ

ðz 2 UÞ:

ð2:27Þ

and

3. Neighborhoods for the multivalently analytic function classes S n;p ðg; k; l; aÞ and Rn;p ðg; k; l; a; m; uÞ In this section, we determine inclusion relations for the classes

S n;p ðg; k; l; aÞ and Rn;p ðg; k; l; a; m; uÞ involving the (n, d)-neighborhoods defined by (1.11) and (1.13). Theorem 3. If

bk = bnþp

ðk ¼ n þ p; n þ p þ 1; n þ p þ 2; . . .Þ

and

d¼

ðn þ pÞðp  aÞwðpÞ ; ðn þ p  aÞwðn þ pÞbnþp

H.M. Srivastava, S. Bulut / Applied Mathematics and Computation 218 (2012) 6511–6518

6517

then d

S n;p ðg; k; l; aÞ  N n;p ðeÞ;

ð3:1Þ

where e is given by (1.12). Proof. The inclusion relation (3.1) would now follow readily from the definition (1.13) and the assertion (2.5) of Lemma 2. h By setting

gðzÞ ¼

z ; 1z

l¼0

m ¼ 2 and

in Theorem 3, we get the following Corollary 4 given by Altıntasß et al. [5]. Corollary 4 [5, Theorem 2]. If f 2 T n ðpÞ is in the class T n ðp; a; kÞ, then d

T n ðp; a; kÞ  N n;p ðeÞ; where e is given by (1.12) and

d¼

ðn þ pÞðp  aÞ½kðp  1Þ þ 1 : ðn þ p  aÞ½kðn þ p  1Þ þ 1

Theorem 4. If

bk = bnþp

ðk ¼ n þ p; n þ p þ 1; n þ p þ 2; . . .Þ

and

! Qm1 ðn þ pÞðp  aÞwðpÞ j¼0 ðu þ j þ pÞ d¼ 1þ ; Q ðn þ p  aÞwðn þ pÞbnþp ðm  1Þ m2 j¼0 ðu þ j þ n þ pÞ then d

Rn;p ðg; k; l; a; m; uÞ  N n;p ðf Þ:

ð3:2Þ

Proof. Suppose that f 2 Rn;p ðg; k; l; a; m; uÞ. Then, upon substituting from (2.16) into the following coefficient inequality: 1 X

kjck  ak j 5

k¼nþp

1 X

kck þ

k¼nþp

1 X

kak

ðck = 0; ak = 0Þ;

k¼nþp

we obtain 1 X

kjck  ak j 5

k¼nþp

1 X k¼nþp

kck þ

Qm1

1 X k¼nþp

j¼0

ðu þ j þ pÞ

j¼0

ðu þ j þ kÞ

Qm1

kck :

ð3:3Þ

Since h 2 S n;p ðg; k; l; aÞ, the assertion (2.5) of Lemma 2 yields

kck 5

ðn þ pÞðp  aÞwðpÞ ðn þ p  aÞwðn þ pÞbnþp

ðk ¼ n þ p; n þ p þ 1; n þ p þ 2; . . .Þ:

ð3:4Þ

Finally, by making use of (2.5) as well as (3.4) on the right-hand side of (3.3), we find that

! Qm1 1 X ðn þ pÞðp  aÞwðpÞ j¼0 ðu þ j þ pÞ ; kjck  ak j 5 1þ Qm1 ðn þ p  aÞwðn þ pÞbnþp k¼nþp k¼nþp j¼0 ðu þ j þ kÞ 1 X

which, by virtue of the sum in (2.21), immediately yields

! Qm1 ðn þ pÞðp  aÞwðpÞ j¼0 ðu þ j þ pÞ ¼: d: kjck  ak j 5  1þ Q ðn þ p  aÞwðn þ pÞbnþp ðm  1Þ m2 k¼nþp j¼0 ðu þ j þ n þ pÞ 1 X

Thus, by applying the definition (1.11), we complete the proof of Theorem 4. h Upon setting

ð3:5Þ

6518

H.M. Srivastava, S. Bulut / Applied Mathematics and Computation 218 (2012) 6511–6518

gðzÞ ¼

z ; 1z

m ¼ 2 and

l¼0

in Theorem 4, we can deduce the following Corollary 5 given by Altıntasß et al. [5]. Corollary 5 [5, Theorem 3]. If f 2 T n ðpÞ is in the class Kn ðp; a; k; uÞ, then d

Kn ðp; a; k; uÞ  N n;p ðf Þ;

ð3:6Þ

where

d¼

ðn þ pÞðp  aÞ½kðp  1Þ þ 1½n þ ðp þ uÞðp þ u þ 2Þ : ðn þ p  aÞ½kðn þ p  1Þ þ 1ðn þ p þ uÞ

Acknowledgements The second-named author was supported by the Kocaeli University (Arslanbey Campus) under Grant HD 2011/22. References [1] O. Altintasß, Neighborhoods of certain p-valently analytic functions with negative coefficients, Appl. Math. Comput. 187 (2007) 47–53. [2] O. Altintasß, On a subclass of certain starlike functions with negative coefficients, Math. Japon. 36 (1991) 489–495. [3] O. Altıntasß, H. Irmak, H.M. Srivastava, Fractional calculus and certain starlike functions with negative coefficients, Comput. Math. Appl. 30 (2) (1995) 9– 15. [4] O. Altintasß, S. Owa, Neighborhoods of certain analytic functions with negative coefficients, Internat. J. Math. Math. Sci. 19 (1996) 797–800. [5] O. Altintasß, Ö. Özkan, H.M. Srivastava, Neighborhoods of a certain family of multivalent functions with negative coefficients, Comput. Math. Appl. 47 (2004) 1667–1672. [6] A.W. Goodman, Univalent Functions, vols. 1, 2, Polygonal Publishing House, Washington, New Jersey, 1983. [7] H. Orhan, M. Kamalı, Fractional calculus and some properties of certain starlike functions with negative coefficients, Appl. Math. Comput. 136 (2003) 269–279. [8] S. Owa, On certain classes of p-valent functions with negative coefficients, Simon Stevin 59 (1985) 385–402. [9] S. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981) 521–527. [10] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975) 109–116. [11] H.M. Srivastava, O. Altıntasß, S. Kirci Serenbay, Coefficient bounds for certain subclasses of starlike functions of complex order, Appl. Math. Lett. 24 (2011) 1359–1363. [12] H.M. Srivastava, S. Owa, S.K. Chatterjea, A note on certain classes of starlike functions, Rend. Sem. Mat. Univ. Padova 77 (1987) 115–124. [13] R. Yamakawa, Certain subclasses of p-valently starlike functions with negative coefficients, in: H.M. Srivastava, S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992, pp. 393–402.