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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 790689, 11 pages

http://dx.doi.org/10.1155/2012/790689

## On Subclasses of Analytic Functions with respect to Symmetrical Points

^{1}Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan^{2}Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan

Received 27 December 2011; Accepted 10 January 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Muhammad Arif et al. 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

In our present investigation, motivated from Noor work, we define the class () of functions of bounded radius rotation of complex order with respect to symmetrical points and learn some of its basic properties. We also apply this concept to define the class . We study some interesting results, including arc length, coefficient difference, and univalence sufficient condition for this class.

#### 1. Introduction

Let denote the class of analytic function satisfying the condition , in the open unit disc and in more simple form: By , , and , we means the well-known subclasses of which consists of univalent, convex, and starlike functions, respectively. In [1], Sakaguchi introduced the class of starlike functions with respect to symmetrical points and is defined as follows: a function given by (1.1) belongs to the class , if and only if Motivated from Sakaguchi work, Das and Singh [2] extend the concepts of to other class in , namely, convex functions with respect to symmetrical points. Let denote the class of convex functions with respect to symmetrical points and satisfying the following condition: Let , , be the class of functions analytic in with and This class was introduced in [3]. For , we obtain the class defined by Pinchuk [4], and for , the class reduces to the class of functions with positive real part.

Now, with the help of the aforementioned concepts, we define the class of functions of bounded radius rotation of complex order with respect to symmetrical points as follows.

*Definition 1.1. *Let in . Then , if and only if
where and .

Using the class , we define the class as follows.

*Definition 1.2. *Let in . Then , if and only if there exists such that
where , , and .

It is noticed that, by giving specific values to , , , and in and , we obtain many well-known as well as new subclasses of analytic and univalent functions; for details see [5–11].

Throughout this paper, we will assume, unless otherwise stated, that , , , and .

Lemma 1.3. *Let be analytic in where belongs to Then
**
(see [8, 12]).*

Lemma 1.4. *Let be univalent function in . Then there exists with such that for all , ,
**
(see [13]).*

#### 2. Some Properties of the Classes

Theorem 2.1. *Let . Then the odd function
**
belongs to in .*

*Proof. *Let and consider
From logarithmic differentiation of the previous relation, we have
or, equivalently,
with
belongs to . Since is a convex set, we have
and hence .

Theorem 2.2. *Let . Then
*

*Proof. *Let . Then by definition we have
Simple computation yields us
Using (2.8) in (2.9), we can easily obtain (2.7).

If we put and in Theorem 2.1, we obtain the integral representation for given by Stankiewiez in [14].

Theorem 2.3. *Let . Then
**
The function defined by
**
shows that this bound is sharp.*

*Proof . *Since , there exists an odd function with
such that
with . Let
Then (2.13) implies that
Equating the coefficients of , we have , and so
where we have used the coefficient bounds for the class .

Corollary 2.4. *The range of every univalent function contains the disc
*

*Proof . *The Koebe one-quarter theorem states that each omitted value of the univalent function of the form (1.1) satisfies
Using (2.18) and Theorem 2.3, we obtain the required result.

By using the same method as in [1], we obtain the following result.

Theorem 2.5. *Let . Then, for and ,
*

Theorem 2.6. *Let . Then, for ,
**
where and
*

*Proof. *We can define, for , , real, the following:
The functions are periodic and continuous with period . Since , therefore from (2.22), it follows that we can choose the branches of argument of and as
Now we have from (2.22)
where is an odd function of the following form:
Since , therefore by using Theorem 2.5, we have
From (2.22), (2.23), (2.24), and (2.27), we have
Moreover, from (2.22)
Therefore

Theorem 2.7. *Let Then for ,
**
where and is a constant depending upon , and only.*

*Proof. *We know that
Since , therefore
By Theorem 2.1, we have, for , the odd function . This implies that
Therefore, we have
Since , therefore we have for odd functions , ,
Now using Cauchy Schwarz inequality, we have
By Lemma 1.3 and distortion results for the class with a subordination result, we obtain
Similarly for we have

Theorem 2.8. *Let . Then for **
where andare the same as in Theorem 2.7 and is a constant depending upon , and only.*

*Proof. *Since, with Cauchy theorem gives
therefore
Now using Theorem 2.7 for we have
Putting we have
Similarly we obtain the required result for .

Theorem 2.9. *Let . Then for ,
*

*Proof. *We know that for and ,
Since , therefore
By Theorem 2.1, we have, for , the odd function . This implies that
Thus, for and , we have
Since , therefore we have for odd functions , ,
By using Lemma 1.4, we have
Now using Cauchy Schwarz inequality, we have
By Lemma 1.3 and distortion result for the class with a subordination result, we obtain
Now putting , we obtain
Similarly for we have

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