International Journal of Mathematics and Mathematical Sciences

Volume 2013, Article ID 294378, 6 pages

http://dx.doi.org/10.1155/2013/294378

## On Certain Classes of Convex Functions

Department of Mathematics, Kyungsung University, Busan 608-736, Republic of Korea

Received 25 February 2013; Accepted 7 May 2013

Academic Editor: Heinrich Begehr

Copyright © 2013 Young Jae Sim and Oh Sang Kwon. 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

For real numbers and such that , we denote by the class of normalized analytic functions which satisfy the following two sided-inequality: where denotes the open unit disk. We find some relationships involving functions in the class . And we estimate the bounds of coefficients and solve the Fekete-Szegö problem for functions in this class. Furthermore, we investigate the bounds of initial coefficients of inverse functions or biunivalent functions.

#### 1. Introduction

Let denote the class of analytic functions in the unit disc which is normalized by Also let denote the subclass of which is composed of functions which are univalent in . And, as usual, we denote by the class of functions in which are convex in .

We say that is subordinate to in , written as , if and only if for some Schwarz function such that If is univalent in , then the subordination is equivalent to

*Definition 1. *Let and be real numbers such that . The function belongs to the class if satisfies the following inequality:

It is clear that . And we remark that, for given real numbers and , if and only if satisfies each of the following two subordination relationships:

Now, we define an analytic function by The above function was introduced by Kuroki and Owa [1], and they proved that maps onto a convex domain conformally. Using this fact and the definition of subordination, we can obtain the following lemma, directly.

Lemma 2. *Let and . Then if and only if
*

And we note that the function , defined by (7), has the form where

For given real numbers and such that , we denote by the class of biunivalent functions consisting of the functions in such that where is the inverse function of .

In our present investigation, we first find some relationships for functions in bounded positive class . And we solve several coefficient problems including Fekete-Szegö problems for functions in the class. Furthermore, we estimate the bounds of initial coefficients of inverse functions and bi-univalent functions. For the coefficient bounds of functions in special subclasses of , the readers may be referred to the works [2–4].

#### 2. Relations Involving Bounds on the Real Parts

In this section, we will find some relations involving the functions in . And the following lemma will be needed in finding the relations.

Lemma 3 (see Miller and Mocanu [5, Theorem ]). *Let be a set in the complex plane and let be a complex number such that . Suppose that a function satisfies the condition
**
for all real and all . If the function defined by is analytic in and if
**
then in . *

Theorem 4. *Let , and
**
Then
*

*Proof. *First of all, we put and note that for . Let
Differentiating (17), we can obtain
where
Using (15), we have
Now for all real with ,
Define a function by
Then is a continuous even function and
Hence and is increasing on , since . Hence satisfies that
for all . Therefore, by combining (21) and (24), we can get
And this shows that for all with . By Lemma 3, we get for all , and this shows that the inequality (16) holds and the proof of Theorem 4 is completed.

Theorem 5. *Let , and
**
Then
*

*Proof. *We put and note that for . And let
As in the proof of Theorem 4, we can get
by (26). And for all real , with ,
where is given by
Since satisfies the inequality
for all . Therefore,
And this shows that for all with . By Lemma 3, we get for all , and this shows that the inequality (27) holds and the proof of Theorem 5 is completed.

By combining Theorems 4 and 5, we can obtain the following result.

Theorem 6. *Let , and
**
Then
*

#### 3. Coefficient Problems Involving Functions in

In the present section, we will solve some coefficient problems involving functions in the class . And our first result on the coefficient estimates involves the function class and the following lemma will be needed.

Lemma 7 (see Rogosinski [6, Theorem 10]). *Let be analytic and univalent in and suppose that maps onto a convex domain. If is analytic in and satisfies the following subordination:
**
then
*

Theorem 8. *Let and be real numbers such that . If the functions , then
**
where is given by
*

*Proof. *Let us define
Then, the subordination (9) can be written as follows:
Note that the function defined by (41) is convex in and has the form
where
If we let
then by Lemma 7, we see that the subordination (42) implies that
where
Now, equality (40) implies that
Then, the coefficients of in both sides lead to
A simple calculation combined with the inequality (46) yields that
where is given by (47) and . Hence, we have . To prove the assertion of the theorem, we need to show that
We now use the mathematical induction for the proof of the theorem. For the case , it is clear. We assume that the inequality (51) holds for . Then, some calculation gives us that
which implies that the inequality (51) is true for . Hence, by the mathematical induction, we prove that
where is given by (47). This completes the proof of Theorem 8.

And now, we will solve the Fekete-Szegö problem for , and we will need the following lemma.

Lemma 9 (see Keogh and Merkes [7]). *Let be a function with positive real part in . Then, for any complex number ,
*

Now, the following result holds for the coefficient of .

Theorem 10. *Let and let the function given by be in the class . Then, for a complex number ,
**
The result is sharp. *

*Proof. *Let us consider a function given by . Then, since , we have , where
where is given by (11). Let
Then is analytic and has positive real part in the open unit disk . We also have
We find from (58) that
which imply that
where
Applying Lemma 9, we can obtain
And substituting
in (62), we can obtain the result as asserted. The estimate is sharp for the function defined by
where the function is given by (7). Hence the proof of Theorem 10 is completed.

Using Theorem 10, we can get the following result.

Corollary 11. *Let and let the function , given by , be in the class . Also let the function , defined by
**
be the inverse of . If
**
then
*

*Proof. *The relations (66) and (67) give
Thus, we can get the estimate for by
immediately. Furthermore, an application of Theorem 10 (with ) gives the estimates for ; hence, the proof of Corollary 11 is completed.

Finally, we will estimate some initial coefficients for the bi-univalent functions .

Theorem 12. *For given and such that , let given by , be in the class . Then
**
where and are given by (63) and (64). *

*Proof. *If , then and , where . Hence
where is given by (7). Let
Then and are analytic and have positive real part in . Also, we have
By suitably comparing coefficients, we get
where and are given by (63) and (64), respectively. Now, considering (76) and (78), we get
Also, from (77), (78), (79), and (80), we find that
Therefore, we have
This gives the bound on as asserted in (71). Now, further computations from (77), (79), (80), and (81) lead to
This equation, together with the well-known estimates
leads us to the inequality (72). Therefore, the proof of Theorem 12 is completed.

#### Acknowledgment

The research was supported by Kyungsung University Research Grants in 2013.

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