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`GeometryVolume 2013 (2013), Article ID 642142, 6 pageshttp://dx.doi.org/10.1155/2013/642142`
Research Article

Fekete-Szegö Coefficient Functional for Certain Subclasses of Close-to-Star Functions

1Department of Mathematics, Khalsa College, Amritsar, Punjab 143001, India
2Department of Mathematics, DIPS College (Co-Educational), Dhilwan, Kapurthala, Punjab 143401, India

Received 8 April 2013; Accepted 10 June 2013

Copyright © 2013 B. S. Mehrok and Gagandeep Singh. 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

We introduce some subclasses of close-to-star functions defined by subordination and obtain sharp upper bounds of the functional , real, for an analytic function , , belonging to these sub-classes.

1. Introduction

Let be the class of bounded functions of the following form: which are analytic in the unit disc and satisfy the conditions and .

It is known (see [1]) that

Let denote the class of functions of the following form: which are analytic in . Let be the class of functions of the form (3) which are analytic univalent in .

We will concentrate on the coefficient problem for the subclasses of . In 1916, Bieberbach [2] proved that for as an elementary corollary area theorem. He conjectured that, for each function , ; the equality holds for the Koebe function , which maps the unit disc onto the entire complex plane minus the slit along the negative real axis from to . De Branges [3] solved the Bieberbach conjecture in 1985. The contribution of Löwner [4] in proving that for the class was huge.

With the known estimates and , it was natural to seek some relation between and for the class . This thought prompted Fekete and Szegö [5] and they used Löwner's method to prove the following well-known result for the class : if , then

The inequality (4) plays a very important role in determining estimates of higher coefficients for some subclasses of (see Chichra [6], Babalola [7]).

Next, we define some subclasses of and obtain the analogy of (4).

We denote by the class of univalent star-like functions which satisfy the following condition:

We denote by the class of convex univalent functions which satisfy the condition

Let and be two analytic functions in . Then, is said to be subordinate to (symbolically ) if there exists a bounded function , such that .

A function is said to be close to star if there exists a function such that

The class of close to star functions was introduced by Reade [8] and is denoted by . A close to star function need not be univalent. The immediate shoot of is the following subclass:

In this paper, we consider the following subclasses of close-to-star functions:

The class was introduced and studied by Mehrok et al. [9]. In particular, , is the class introduced and studied by Mehrok et al. [10]. Obviously, .

Hummel [11] proved the conjecture of . Singh that for the class . Keogh and Merkes [12] obtained the estimates (4) for the classes and . We obtain sharp upper bounds of the functional for functions belonging to the classes and .

2. Preliminary Lemmas

Lemma 1. Let . Then,

This lemma is due to Keogh and Merkes [12].

Lemma 2. Let . Then,

This result was proved by Keogh and Merkes [12].

3. Main Results

Theorem 3. Let . Then,

These results are sharp.

Proof. By definition of ,
Identifying terms in the above expansion,
From (15) and by using (2), it is easily established that where and .
Two cases arise: (a) and (b) .
Consider the case .
Case I. Suppose that so that .
By Lemma 1, (16) can be written as implies that .
Subcase I(i). For ,  provided .
Where   is an increasing function in and max.
Subcase I(ii). Suppose and .
If and .
Combining the above two subcases, we obtain the first result of (13). For which holds for .
For or (for all and ), from which the second result of (13) follows.
We now consider the case .
Case II. Suppose that . Then by Lemma 1, (16) takes the following form:
Subcase II(i). For , where implies that .
For , we have and . So , which gives the third result of (13).
Subcase II(ii). For . For ,, and , which leads us to the fourth result of (13).
Case III. For . By Lemma 1, (16) can be put in the following form: where shows that .
If ,  then , which is true.
for . So , which takes us straight to the fifth result of (13).
The equality holds in the first and fifth results of (13) for the function defined by
The equality holds in the second result of (13) for the function defined by where .
The equality holds in the third result of (13) for the function defined by where .
The equality holds in the fourth result of (13) for the function defined by where and for .
Proof of Theorem 3 is completed.

For and , Theorem 3 gives the following result.

Corollary 4. If , then,

Theorem 5. Let . Then,

The results are sharp.

Proof. Proceeding as in Theorem 3, we have where and .
Two cases arise: (a) ,  (b) .
Consider the case .
By Lemma 2, (27) can be written as
Subcase I(i). For ,
For , which shows that is an increasing function in and max.
Subcase I(ii). Suppose .
implies that .
For . So and max.
Combining the above two subcases, we obtain the first result of (26).
For , which holds for .
So , which leads us to the second result of (26).
We now consider the case .
Subcase I(iii). For , (27) takes the following form: where implies that .
It is easy to verify that . So max, which gives the third result of (26).
Case II. For .
Subcase II(i). For , (27) reduces to where implies that .
For , , , and , which leads us to the fourth result of (26).
Subcase II(ii). For , (27) reduces to the following form: implies that . .
Combining the above two subcases, , which take us straight to the fifth result of (26).
Case III. For . By Lemma 1, (27) can be put in the form where shows that .
and , which yield the sixth result of (26).
The equality holds in the first and sixth results of (26) for the function defined by
The equality holds in the second result of (26) for the function defined by where .
The equality holds in the third and fourth results of (26) for the function defined by where .
The equality holds in the fifth result of (26) for the function defined by where and for .
Proof of Theorem 5 is completed.

For , Theorem 5 gives the following result.

Corollary 6. If , then

The results of Corollaries 4 and 6 were proved by the authors of [13].

References

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