Abstract and Applied Analysis

Volume 2011, Article ID 361647, 15 pages

http://dx.doi.org/10.1155/2011/361647

## Some Properties of Subclasses of Multivalent Functions

Department of Mathematics, Faculty of Science, Atatürk University, 25240 Erzurum, Turkey

Received 8 November 2010; Accepted 8 January 2011

Academic Editor: Ondřej Došlý

Copyright © 2011 Muhammet Kamali and Fatma Sağsöz. 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

The authors introduce two new subclasses denoted by and of the class of -valent analytic functions. They obtain coefficient inequality for the class . They investigate various properties of classes and . Furthermore, they derive partial sums associated with the class .

#### 1. Introduction and Definition

Let denote the class of functions of the form
which are analytic and *p*-valent in the open unit disc . We write .

A function is said to be in the class of *p-*valently star-like functions of order if it satisfies the condition
Furthermore, a function is said to be in the class of *p-*valently convex functions of order if it satisfies the condition
The classes and were studied by Owa [1]. The class was considered by Patil and Thakare [2].

We denote by the subclass of the class consisting of functions of the form and define two further classes and by For the classes .

The following lemmas were given by Owa [3].

Lemma 1.1. *Let the function be defined by
**
Then, is in the class if and only if
**
The result is sharp.*

Lemma 1.2. *Let the function be defined by (1.6). Then, is in the class if and only if
**
The result is sharp.*

For a function defined by (1.6) and in the class , Lemma 1.1 yields
On the other hand, for a function defined by (1.6) and in the class , Lemma 1.2 yields
In view of the coefficient inequalities (1.9) and (1.10), it would seem to be natural to introduce and study here two further classes and of analytic and *p*-valent functions, where denotes the subclass of consisting of functions of the form
and denotes the subclass of consisting of functions of the form
The classes and are studied by Aouf et al. [4].

The classes

were considered earlier by Silverman and Silvia [5].

Now, we give the following equalities for the functions belonging to the class : We define such that

A function is said to be in the class if it satisfies the inequality for some , and for all .

If and , we obtain the condition (1.2). Furthermore, we obtain the condition (1.3) for and .

We denote by the subclass of the class consisting of functions of the form and define the class by

Furthermore, we denote by the subclass of consisting of functions of the form The main object of the present paper is to investigate interesting properties and characteristics of the classes and . Also, the partial sums is defined for function defined by (1.19).

#### 2. A Coefficient Inequality for the Class and Some Theorems for the Class

First, we give a coefficient inequality for the class .

Theorem 2.1. *Let the function be defined by (1.17). Then, is in the class if and only if
*

*Proof. *Suppose that . Then, we find from (1.16) that
If we choose to be real and let , we get
or, equivalently,
Thus, we have
or
Conversely, suppose that the inequality (2.1) holds true and let
Then, we find from the definition (1.4) that
By means of inequality (2.1), we can write
or
Thus, we obtain
This evidently completes the proof of Theorem 2.1.

Now, we give a characterization theorem for the class .

Theorem 2.2. *Let the function be defined by (1.19). Then, is in the class if and only if
**
The result is sharp for the function given by
*

*Proof. *Using inequality (2.1), we have
or
Thus, by setting
we obtain
or
A closure theorem for the class is given by the following.

Theorem 2.3. *Let
**
If , then the function given by
**
with
**
is also in the class .*

*Proof. *Since for , it follows from Theorem 2.2 that
By applying (2.22) and the definition (2.21), we write

Theorem 2.4. *Let
**
where . **Then, is in the class if and only if it can be expressed in the form
*

*Proof. *Suppose that is given by (2.25), so that we find from (2.24) that
where the coefficients are given with , . Then, since,
we conclude from Theorem 2.2 that .

Conversely, let us assume that the function defined by (1.19) is in the class . Then,
which follows readily from (2.12) for .

For , setting
and , we thus arrive at (2.25). This completes the proof of Theorem 2.4.

Theorem 2.5. *Let be given by (1.19) and define the partial sums by
**
Suppose also that
**
Then, for , one has
**
Each of the bounds in (2.32) and (2.33) is the best possible.*

*Proof. *From (2.31) and Theorem 2.2, we have that . By definition , we can write
Under the hypothesis of this theorem, we can see from (2.31) that , for . Therefore, we have
Using (1.19) and (2.30), we can write
Set
By applying (2.35) and (2.37), we find that
which shows that . Thus, we obtain
or
Let
Then, satisfies the condition (2.31) and . Thus, we can write
and taking as
which shows that the bound in (2.32) is the best possible.

By using definitions of and , we can write
If we put
and make use of (2.35), we can deduce that
requires that . Thus, we obtain
or
It follows from the last inequality that assertion (2.33) of the Theorem 2.5 holds.

The bound in (2.33) is sharp with the extremal function given by (2.41). The proof of the theorem is thus completed.

If , , and are taken in Theorem 2.5, the following result is obtained given by Liu [6].

Corollary 2.6. *Let be given by (1.19) and define the partial sums by
**
Suppose also that
**
Then, for , one has
**
Each of the bounds in (2.51) is the best possible.*

#### References

- S. Owa, “Some properties of certain multivalent functions,”
*Applied Mathematics Letters*, vol. 4, no. 5, pp. 79–83, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. A. Patil and N. K. Thakare, “On convex hulls and extreme points of $p$-valent starlike and convex classes with applications,”
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*Simon Stevin*, vol. 59, no. 4, pp. 385–402, 1985. View at Google Scholar - M. K. Aouf, H. M. Hossen, and H. M. Srivastava, “Some families of multivalent functions,”
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