#### 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).