International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 362312, 21 pages

doi:10.1155/2012/362312

Research Article

Properties of Certain Subclass of Multivalent Functions with Negative Coefficients

Jingyu Yang1,2 and Shuhai Li¹

¹Department of Mathematics, Chifeng University, Inner Mongolia, Chifeng 024000, China

²School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, China

Received 26 December 2011; Revised 26 March 2012; Accepted 27 March 2012

Academic Editor: Marianna Shubov

Copyright © 2012 Jingyu Yang and Shuhai Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Making use of a linear operator, which is defined by the Hadamard product, we introduce and study a subclass of the class . In this paper, we obtain the coefficient inequality, distortion theorem, radius of convexity and starlikeness, neighborhood property, modified convolution properties of this class. Furthermore, an application of fractional calculus operator is given. The results are presented here would provide extensions of some earlier works. Several new results are also obtained.

1. Introduction

Let denote the class of functions

which are analytic and -valent in the open unit disc on the complex plane .

We denote by and the subclass of -valent starlike functions and the subclass of -valent convex functions, respectively, that is, (see for details [1, 2])

A function is said to be analytic starlike of order if it satisfies

for some and for all .

Further, a function is said to be analytic convex of order if it satisfies

for some and for all .

For and , the Hadamard product (or convolution) is defined by

The linear operator is defined as follows (see Saitoh [3]):

and is defined by

where is the pochharmmer symbol defined (in terms of the Gamma function) by

The operator was studied recently by Srivastava and Patel [4]. It is easily verified from (1.6) and (1.7) that

Moreover, for ,

where denotes the Ruscheweyh derivative of a function of order (see [5]).

Aouf, Silverman and Srivastava [6] introduced the class of by use of linear operator , further investigated the properties of this class. In [7], SokóŁ investigated several properties of the linear Aouf-Silverman-Srivastava operator and furthermore obtained the corresponding characterizations of multivalent analytic functions which were studied by Aouf et al. in paper [6]. The properties of multivalent functions with negative coefficients were studied in [8–10]. References [11, 12] gave the results of the univalent function with negative coefficients.

In this paper, we will use operator to define a new subclass of as follows.

For and for the parameters such that

we say that a function is in the class if it satisfies the following subordination:

or equivalently, if the following inequality holds true:

From the above definition, we can imply that the function class in [6] is the special case of in our present paper because .

Since , then, like [6], we have the following subclasses which were studied in many earlier works:

(1)(Aouf and Darwish [8]),(2)(Shukla and Dashrath [9]),(3)(Lee et al. [10]),(4)(Gupta and Jain [11]),(5) (Uralegaddi and Sarangi [12]).

The purpose of this paper is to give various properties of class . We extend the results of basic paper [6].

2. Necessary and Sufficient Condition for

Theorem 2.1. Let the function be given by (1.1), then if and only if

where

Proof. For the sufficient condition, let , we have

then

By the maximum modulus theorem, for any , we have

this implies .

For necessary condition, let be given by (1.1), then

from (1.13) and (2.6), we find that

Now, since for all , we have

We choose value of on the real axis so that the expression is real.

Then, we have

Letting throughout real values in (2.10), we get

So we have

it is

This completes the proof of the theorem.

Corollary 2.2. Let the function be given by (1.1). If then

The result is sharp for the function given by

3. Distortion Inequality of Class

Theorem 3.1. If a function defined by (1.1) is in the class , then

The result is sharp for the function given by

Proof. Since , then

From , using Theorem 2.1, we have

which readily yields

By (3.3) and (3.5), we can imply (3.1).

4. Radius of Starlikeness and Convexity

Theorem 4.1. Let the function defined by (1.1) be in the class . Then, is starlike of in where

and is defined by (2.1).

Proof. We must show that

Since

to prove (4.1), it is sufficient to prove

It is equivalent to

By Theorem 2.1, we have

hence (4.5) will be true if

It is equivalent to

This completes the proof.

Theorem 4.2. Let the function defined by (1.1) be in the class . Then, is convex of in where

and is defined by (2.1).

Proof. It sufficient to show that

Since

to prove (4.9), it is sufficient to prove

It is equivalent to

By Theorem 2.1, we have

hence (4.13) will be true if

It is equivalent to

This completes the proof.

5. δ-Neighborhood of

Based on the earlier works by Aouf et al. [6], Altintas et al. [13–15], and Aouf [16], we introduce the -neighborhood of a function of the form (1.1) and present the relationship between -neighborhood and corresponding function class.

Definition 5.1. For , the -neighborhood of a function is defined as follows:

where

Theorem 5.2. Let the function defined by (1.1) be in the class . Then,

This result is the best possible in the sense that cannot be increased.

Proof. Let , then by Theorem 2.1, we have

which is equivalent to

For any

we find from (5.1) that

By (5.5) and (5.7), we get

By Theorem 2.1, it implies that .

To show the sharpness of the assertion of Theorem 5.2, we consider the functions and given by

where

Since

so .

But

by Theorem 2.1.

6. Properties Associated with Modified Hadamard Product

Following early works by Aouf et al. in [6], we provide the properties of modified Hadamard product of .

For the function

the modified Hadamard product of the functions and was denoted by and defined as follows:

Theorem 6.1. Let given by (6.1) be in the class , then

where

The result is sharp for the functions given by

Proof. By Theorem 2.1, we need to find the largest such that

Since , then we see that

By Cauchy-Schwartz inequality, we obtain

This implies that we only need to show that

or equivalently that

By making use of the inequality (6.8), it is sufficient to prove that

From (6.11), we have

Define the function by

We see that is an increasing function of . Therefore, we conclude that

By using arguments similar to those in the proof of Theorem 6.1, we can derive the following result.

Theorem 6.2. Let defined by (6.1) be in the class , defined by (6.1) be in the class , then

where

The result is sharp for the functions given by

Theorem 6.3. Let defined by (6.1) be in the class , then the function defined by

belongs to the class where

This result is sharp for the functions given by (6.5).

Proof. By Theorem 2.1, we want to find the largest such that

Since , we readily see that

From (6.21), we have

then we have

From (6.23), if we want to prove (6.20), it is sufficient to prove there exists the largest such that

that is

Now we define by

We see that is an increasing function of . Therefore, we conclude that

which completes the proof of Theorem 6.1.

7. Application of Fractional Calculus Operator

References [17–19] have studied the fractional calculus operators extensively. In this part, we only investigate the application of fractional calculus operator which was defined by [6] in the class of .

Definition 7.1 (see [6]). The fractional integral of order is defined, for a function , by

where the function is analytic in a simply connected domain of the complex z-plane containing the origin and the multiplicity of is removed by requiring to be real when .

Definition 7.2 (see [6]). The fractional integral of order is defined, for a function , by

where the function is constrained, the multiplicity of is removed as in Definition 7.1.

In our investigation, we will use the operators defined by (cf, [20–22])

as well as for which it is well known that (see [23])

in terms of Gamma.

Lemma 7.3 (see Chen et al. [18]). For a function ,

provided that no zeros appear in the denominators in (7.5).

Remark 7.4. Throughout this section, we assume further that .

Theorem 7.5. Let function defined by (1.1) be in the class , then

, each of the assertions is sharp.

Proof. Since

then

Let is defined by

where

Since is a decreasing function of when , we get

since , by Theorem 2.1, we get

It is that

then

From (7.15), we obtain (7.6) and (7.7), respectively.

Equalities in (7.6) and (7.7) are attained for the function given by

Theorem 7.6. Let function defined by (1.1) be in the class , then

, each of the assertions is sharp.

Proof. Since

then

Let be defined by

where

Since is a decreasing function of when , then

since , by Theorem 2.1, we get

It is that

then

From (7.26), we obtain (7.17) and (7.18), respectively.

Equalities in (7.17) and (7.18) are attained for the function given by

or equivalently by

Acknowledgments

The author thanks the referee for numerous suggestions that helped make this paper more readable. The present investigation was partly supported by the Natural Science Foundation of Inner Mongolia under Grant 2009MS0113.

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