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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 307526, 9 pages
General Univalence Criterion Associated with the th Derivative
1Department of Mathematics, Faculty of Science and Information Technology, Zarqa University, Zarqa 13132, Jordan
2School of Mathematical Sciences, Universiti Kebangsaan Malaysia, Selangor, 43600 Bangi, Malaysia
Received 23 February 2012; Revised 2 May 2012; Accepted 10 May 2012
Academic Editor: Saminathan Ponnusamy
Copyright © 2012 Oqlah Al-Refai and Maslina Darus. 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.
For normalized analytic functions with for , we introduce a univalence criterion defined by sharp inequality associated with the th derivative of , where .
Let denote the class of functions of the following form: which are normalized analytic in the open unit disk .
In , Aksentev proved that the condition or equivalently , for , is sufficient for to be univalent in . By virtue of the aforementioned result of Aksentev, the class of functions defined by (1.2) was extensively studied by Obradović and Ponnusamy [2, 3], Ozaki and Nunokawa , Obradović et al. , and others. Afterwards, Nunokawa et al.  proved for with when that implies for , and hence is univalent in . Later, Yang and Liu  extended this result for : with when implies that is univalent in and the bound is best possible for univalence. This result was also given first in the preprint of reports of the Department of Mathematics, University of Helsinki: M. Obradović, S. Ponnusamy, New criteria, and distortion theorems for univalent functions, Preprint 190, June 1998. Later, under the same name, the paper was published in Complex Variables Theory Application (see ). Corresponding to the functions defined by (1.4), Yang and Liu in  studied a class of analytic univalent functions satisfying and denoted by . The class is extensively studied in the recent years (see [2, 3, 8–10]).
In this work, we introduce a univalence criteria defined by the conditions for and where is normalized analytic in and , . The sharpness occurs for the Koebe function. Indeed, all functions satisfying the condition (1.5) are univalent in and the bound 1 in the inequality is best possible for univalence. Letting in (1.5) gives the univalence criterion defined by (1.4). Some special cases and examples for functions satisfying (1.5) are given.
2. Sufficient Conditions for Univalence
Let us prove the following theorem.
Theorem 2.1. Let with for and let be bounded in and satisfy For any , if where and , then is univalent in .
Proof. If we put then the function is analytic in and, by integration from to , we get Integrating both sides of the previous equation -times from to gives Thus, we have where Next, for , we have and for , In general, for , therefore and so for and . If , then , and it follows, from (2.7) and (2.13), that Hence, is univalent in .
Corollary 2.2. Let with when . For any , if where , then is univalent in . The result is sharp, where equality occurs for the Koebe function and also for functions of the following form:
Proof. Setting in Theorem 2.1 immediately yields (2.15). To show that the result is sharp for , we consider A computation shows, for , that Letting in (2.17) and (2.18) implies, respectively, that and This satisfies the equality in (2.15), because for and , an application of the binomial theorem gives and so Choosing in assertion (2.21) gives the equality. However, for every , we have Hence is not univalent in and the result is sharp. Moreover it can be easily checked that the equality in (2.15) holds for the given functions and the proof is complete.
3. Special Cases and Examples
Letting in inequality (2.15) gives the univalence criterion defined by (1.4), which is due to Yang and Liu . Next, we reduce the result for some values of by computing the corresponding values of in terms of the coefficients. More precisely, for and , Corollary 2.2 reduces at once to the following two remarks.
Remark 3.1. Let with when satisfy Then is univalent in . The bound in (3.1) is best possible, where equality occurs for the Koebe function and for functions of the following form:
Proof. The result follows from taking in Corollary 2.2 and that .
Remark 3.2. Let with for satisfy Then is univalent in . The bound in (3.3) is best possible, where equality occurs for the Koebe function and also for functions of the following form:
Proof. The result follows from taking in Corollary 2.2 and that , and .
To understand the behavior of the extremal functions for our criterion (2.15), let us consider, for example, , which is an extremal function for the case . Figures 1(a) and 1(b) show the images of the unit circle under the functions and , respectively. If we restrict the images around the cusps as shown in Figures 1(c) and 1(d), we see that the image of is a curve that intersects itself in some purely real point . This means that there are two different points and that lie on the unit circle such that . In fact, each purely real point lies inside the closed curve of Figures 1(c) and 1(d) which is an image for two different points in having the same modulus but different arguments. However, we cannot find such points for the function , and this interprets why is an extremal function for univalence, since the closed curve of Figure 1(d) vanishes whenever the power in the function approaches to as shown in Figure 1(c).
From Corollary 2.2, we have the following.
Corollary 3.3. Let with for and for some . Then is univalent in .
Example 3.5. From Corollary 3.3, it can be easily seen that the functions with for and , are univalent in .
The authors gratefully thank the referees for their remarkable comments and gratefully acknowledge the financial support received in the form of the Research Grant UKM-ST-06-FRGS0244-2010 from the Universiti Kebangsaan Malaysia.
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