Research Article | Open Access

# New Subclasses of Analytic Functions with Respect to Symmetric and Conjugate Points

**Academic Editor:**Jacek Dziok

#### Abstract

We introduce new subclasses of close-to-convex and quasiconvex functions with respect to symmetric and conjugate points. The coefficient estimates for functions belonging to these classes are obtained.

#### 1. Introduction

Let be the class of functions which are analytic and univalent in the open unit disk given by and satisfying the conditions , , .

Let denote the class of functions which are analytic and univalent in of the form

Let be the subclass of functions and satisfying the condition These functions are called starlike with respect to symmetric points and were introduced by Sakaguchi [1].

Also, let be the subclass of functions and satisfying the condition These functions are called starlike with respect to conjugate points and were introduced by El-Ashwah and Thomas [2]. Further results on starlike functions with respect to symmetric points or conjugate points can be found in [3–5].

Then, Das and Singh [6] introduced another class , namely, convex functions with respect to symmetric points and satisfying the condition

Suppose that and are two analytic functions in . Then, we say that the function is subordinate to the function , and we write , , if there exists a Schwarz function with and such that .

In view of subordination definition, Goel and Mehrok [7] introduced a subclass of denoted by .

Let be the class of functions of the form (2) and satisfying the condition

Following them, many authors introduced the analogue definitions by extension as follows (see [8, 9]).

*Definition 1. *Let be the subclass of consisting of functions given by (2) satisfying the condition

Let be the subclass of consisting of functions given by (2) satisfying the condition

Let be the subclass of consisting of functions given by (2) satisfying the condition

Motivated by the pervious classes, Selvaraj and Vasanthi [10] defined the following classes of functions with respect to symmetric and conjugate points.

*Definition 2. *Let be the subclass of consisting of functions given by (2) satisfying the condition

Let be the subclass of consisting of functions given by (2) satisfying the condition

In this paper, we introduce the class consisting of analytic functions of the form (2) and satisfying where .

In addition, we introduce the class consisting of analytic functions of the form (2) and satisfying where .

We note that(i)for , (see Mehrok et al. [11]) and ;(ii)for and , (see Janteng and Halim [12]) and ;(iii)for and , and ;(iv)for , and ;(v)for and , (see Janteng and Halim [13]) and ;(vi)for and , and .

By the definition of subordination, it follows that if and only if and that if and only if where

In the next section, we discuss the coefficient estimates for functions belonging to the classes and .

#### 2. Some Preliminary Lemmas

We will require the following lemmas for proving our main results.

Lemma 3 (see [7]). * If is given by (14), (15) and (16), then for ,
*

Lemma 4 (see [10]). * Let . Then, for , ,
*

Lemma 5 (see [10]). * Let . Then, for , ,
*

#### 3. Main Results

Unless otherwise mentioned, we will assume in the reminder of this paper that , , and .

Theorem 6. * Let , then for ,
*

*Proof. *Since , it follows that
for , with , where .

On equating the coefficients of like powers of in (22), we get
and continuing in this way, we obtain
From (14) and (16), we have

On equating the coefficients, we obtain
and so
By using Lemma 3 and (27), we have
Again, by applying Lemma 3 and using (23), we obtain from (28)
It follows that (20) and (21) hold for . We now prove (20) and (21) by induction.

Equation (29) together with Lemma 3 yield

Again, using Lemma 3 in (25), we have

Using (34) in (33), we obtain
We suppose that (20) and (21) hold for .

Using Lemma 4 in (32) and (35), we get
In order to prove (20), it is sufficient to show that
Thus, (38) is valid for .

Let us assume that (38) is true for all . Then, from (36), we have
Thus, (38) holds for , and, hence, (20) follows. Next, we prove (21).

From (38), we have

By using (40) in (37), we obtain
which proves (21).

By taking and in Theorem 6, respectively, we can readily deduce the following corollaries.

Corollary 7 (see [11]). * Let , then, for ,
*

Corollary 8. * Let , then, for ,
**
Further, putting and in Corollary 8, one obtains the following.*

Corollary 9. * Let , then, for ,
*

*Remark 10. * Corollary 9 improves the result obtained by Janteng and Halim [13, Theorem 3.1].

Theorem 11. * Let , then, for ,
*

*Proof. * Since it follows that
where .

On equating the coefficients of like powers of in (47), we get
and continuing in this way, we obtain
From (15) and (16), we have

On equating the coefficients, we obtain
and so
By using Lemma 3, (48), and (54), we have
Again, by applying Lemma 3 and using (48)–(50), we obtain from (55) and (56)
It follows that (45) and (46) hold for . We now prove (45) by induction.

Equation (57) together with Lemma 3 yield
Again, by using Lemma 3 in (51), we have
Using (62) in (61), we obtain
We suppose that (45) holds for .

Using Lemma 5 in (63), we get
In order to prove (45), it is sufficient to show that
Thus, (65) is valid for .

Let us assume that (65) is true for all , . Then, from (64), we have
Thus, (65) holds for , and, hence, (45) follows. Similarly, we can prove (46).

By taking and in Theorem 11, respectively, we can easily derive the following corollaries.

Corollary 12. * Let , then, for ,
*

Corollary 13. * Let , then, for ,
*

#### Acknowledgments

The present investigation was partly supported by the Natural Science Foundation of China under Grant no. 11271045, the Higher School Doctoral Foundation of China under Grant no. 20100003110004, and the Natural Science Foundation of Inner Mongolia of China under Grant no. 2010MS0117. The authors would like to thank Professor Jacek Dziok for his valuable suggestions and the referees for their careful reading and helpful comments to improve the paper.

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#### Copyright

Copyright © 2013 Huo Tang and Guan-Tie Deng. 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.