Criteria for Strongly Starlike Functions

Abstract

Let be analytic in the unit disk with and . By using the method of differential subordinations, we determine the largest number such that, for some and , the differential subordination implies Some useful consequences of this result are also given.

1. Introduction

Let be the class of functions of the form which are analytic in . A function is said to be starlike of order if for some . We denote this class by . A function is said to be convex of order if for some . We denote this class by . Further, a function is said to be strongly starlike of order if for some . Also we denote this class by . Clearly for .

Let and be analytic in . Then the function is said to be subordinate to , written as , if there exists an analytic function with and such that for . If is univalent in , then is equivalent to and . It is easy to see that a function if and only if

A number of results for strongly starlike functions in have been obtained by several authors (see, e.g., [1–7]). In this paper, by using the method of differential subordinations, we determine the largest number such that, for some , and , the differential subordination implies Our results improved or extended the above results.

To prove our results, we need the following lemma due to Miller and Mocanu [3].

Lemma 1.1. Let be analytic and univalent in U, and let and be analytic in a domain D containing , with when . Set and suppose that (i) is starlike univalent in U, (ii).
If is analytic in U, with , then and is the best dominant of (1.9).

2. Main Results

Theorem 2.1. Let an integer, , and . If satisfies and where then and given by (2.2) is the largest number such that (2.3) holds.

Proof. Let with , and define the function in by Then is analytic in and Let an integer, , and and choose Then is analytic and univalent in and satisfy the conditions of the lemma. The function is univalent and starlike in because for . Further, we have that Since , we have that and so Inequality (2.13) shows that the function is close-to-convex and univalent in . Letting and , then and so where It is easy to know that takes its minimum value at . Hence, in view of , we deduce that where is given by (2.2).
Now, if satisfies (2.1), it follows from (2.17) that the subordination holds. Hence it follows from (2.5), (2.7), (2.10), and (2.18) that holds. Therefore, by virtue of the lemma, we conclude that , that is, (2.3) holds.
Next we consider the extremal function Then satisfies and it follows from (2.17) that the bound in (2.1) is sharp. The proof of the theorem is complete.

Making use of the theorem, we can obtain a number of interesting results.

Letting and in the theorem, we have the following corollary.

Corollary 2.2. Let and . If satisfies and where then and given by (2.23) is the largest number such that (2.24) holds.

Remark 2.3. Ramesha et al. [6] have proved that, if satisfies and then .
For and , it follows from (2.23) that . Hence, the image of is a region which properly contains the right half plane. Thus we conclude that Corollary 2.2 with and is better than the result of Ramesha et al. [6].
Letting in Corollary 2.2, we have the following corollary.

Corollary 2.4. Let . If satisfies and then , where is the root of the equation

Remark 2.5. For , Corollary 2.4 reduces to a main result of Nunokawa et al. [5, Theorem 1] by a different method.

Remark 2.6. Note that . Hence it follows from Corollary 2.2 that implies
Letting in the theorem, we have the following corollary.

Corollary 2.7. Let . If satisfies and where then and given by (2.31) is the largest number such that (2.32) holds.

Remark 2.8. Marjono and Thomas [2] proved the above result by a different method. For Corollary 2.2 reduces to a result of Darus [1]. For , Corollary 2.2 reduces to a result of Nunokawa and Thomas [4].
Letting and in the theorem, we have the following corollary.

Corollary 2.9. Let and . If satisfies and where then and given by (2.34) is the largest number such that (2.35) holds.

Remark 2.10. For , Corollary 2.9 reduces to a result of Ravichandran and Darus [7].

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant no. 10871094) and Natural Science Foundation of Universities of Jiangsu Province (Grant no. 08KJB110001).

References

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Copyright

Copyright © 2011 Neng Xu. 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.