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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 312387, 8 pages
http://dx.doi.org/10.1155/2013/312387
Research Article

On Certain Subclass of Analytic Functions with Fixed Point

1School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia
2Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 130095, Jordan

Received 14 October 2012; Revised 27 December 2012; Accepted 27 December 2012

Academic Editor: Yongkun Li

Copyright © 2013 T. Al-Hawary 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

A new certain differential operator and a subclass are introduced for functions of the form which are univalent in the unit disc . In this paper, we obtain coefficient inequalities, distortion theorem, closure theorems, and class preserving integral operators of functions belonging to the class .

1. Introduction

Let be the set of functions which are regular in the unit disc , and is univalent in .

We recall here the definitions of the well-known classes of starlike and convex functions:

Let be a fixed point in and .

In [1], Kanas and Rønning introduced the following classes: These classes are extensively studied by Acu and Owa [2].

The class is defined by geometric property that the image of any circular arc centered at is starlike with respect to and the corresponding class is defined by the property that the image of any circular arc centered at is convex. We observe that the definitions are somewhat similar to the ones for uniformly starlike and convex functions introduced by Goodman in [3, 4], except that in this case the point is fixed.

In fact, several subclasses of have been introduced by a fixed geometric property of the image domain. It is interesting to note that these subclasses play an important role in other branches of mathematics. For example, starlike functions and convex functions play an important role in the solution of certain differential equations (see Robertson [5], Saitoh [6], Owa et al. [7], Reade and Silverman [8], and Sokół and Wiśniowska-Wajnryb [9]). One of the important problems in geometric function theory are the extremal problems, which pose an effective method for establishing the existence of analytic functions with certain natural properties. Extremal problems (supremum ) play an important role in geometric function theory, for finding sharp estimates, coefficient bounds, and an extremal function. The results we obtained here may have potential application in other branches of mathematics, both pure and applied. For example the extremal problems are closely connected to Hele-Shaw flow of fluid mechanics [10], exterior inverse problems in potential theory, and many others.

Now let us begin with our definitions as follows.

The function in is said to be starlike functions of order if and only if for some . We denote by the class of all starlike functions of order . Similarly, a function in is said to be convex of order if and only if for some . We denote by the class of all convex functions of order .

It is easy to see that denoting the subclass of has the series of expansion:

For the function in the class , we define the following new differential operator: and for . for , , , , and .

It easily verified from (7) that

Remark 1. (i) When , we have the operator introduced and studied by Darus and Ibrahim (see [11]).
(ii) When and , we have the operator introduced and studied by Al-Oboudi (see [12]).
(iii) When and , we have the operator introduced and studied by Acu and Owa (see [2]).
(iv) And when and , we have the operator introduced and studied by Sǎlǎgean (see [13]).

With the help of the differential operator , we define the class as follows.

Definition 2. A function is said to be a member of the class if it satisfies for some , , , , , and and for all .

It is easy to check that is the class of starlike functions of order and gives the all class of starlike functions.

Let denote the subclass of consisting of functions of the following form:

Further, we define the class by

In this paper, coefficient inequalities, distortion theorem, and closure theorems of functions belonging to the class are obtained. Finally, the class preserving integral operators of the form for the class is considered.

2. Coefficient Inequalities

Our first result provides a sufficient condition for a function, regular in , to be in .

Theorem 3. A function being defined by (10) is in if and only if where . The result (13) is sharp for functions of the following form:

Proof. Suppose that (13) holds true for . consider the expression Then for , we have
So that .
 For the converse, assume that Since for all , it follows from (17) that
Choose values of on the real axis so that is real. Upon clearing the denominator in (18) and letting through real values, we obtain This gives the required condition. Hence the theorem follows.

Corollary 4. Let the function be defined by (10) and . If , then
The result (20) is sharp for functions given by (14).

3. Distortion Theorem

A distortion property for functions in the class is contained in

Theorem 5. Let the function one has defined by (10) be in the class . Then for , we have

Proof. Since , Theorem 3 readily yields the inequality Thus, for , and making use of (22), we have Also from Theorem 3, it follows that Hence This completes the proof of Theorem 5.

4. Radii of Starlikeness and Convexity

The radii of starlikeness and convexity for the class is given by the following theorems.

Theorem 6. If the function defined by (10) is in the class , then is of order in , where
The result is sharp for the functions given by (14).

Proof. It suffices to prove that for . We have Hence (28) holds true if or with the aid of (13), (30) is true if Solving (31) for , we obtain This completes the proof of Theorem 6.

Theorem 7. If the function defined by (10) is in the class , then is convex of order in , where
The result is sharp for the functions given by (14).

Proof. By using the technique employed in the proof of Theorem 6, we can show that for , with the aid of Theorem 3. Thus we have the assertion of Theorem 7.

5. Closure Theorems

Let the functions ,, be defined by for .

Theorem 8. Let the functions be defined by (35) in the class for every . Then the function defined by is a member of the class , where

Proof. Since , it follows from Theorem 3 that for every . Hence, which (in view of Theorem 5) implies that .

Theorem 9. The class is closed under convex linear combination.

Proof. Suppose that the functions defined by (35) be in the class , it is sufficient to prove that the function is also in the class . Since, for , we observe that Hence . This completes the proof of Theorem 9.

Theorem 10. Let Then if and only if it can be expressed in the form where and .

Proof. Suppose that where    and . Then Since it follows from Theorem 3 that the function . Conversely, let us suppose that . Since Setting it follows that . This completes the proof of the theorem.

6. Extreme Points

Theorem 11. Let Then if and only if it can be expressed in the form where and .

Proof. Suppose that where    and . Then Since it follows from Theorem 3 that the function . Conversely, let us suppose that . Since Setting it follows that . This completes the proof of theorem.

7. Integral Operators

Theorem 12. If the function defined by (10) is in the class , and let be a real number such that . Then the function defined by also belongs to the class .

Proof. From (57), it follows that , where . Therefore since . Hence by Theorem 3, .

Acknowledgment

The work here was fully supported by LRGS/TD/2011/UKM/ICT/03/02 and UKM-DLP-2011-050.

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