Abstract

Let be the class of functions of the form which are analytic in . By using the method of differential subordinations, we give some sufficient conditions for Carathéodory function, that is, if satisfies and where , , , ,, then . Some useful consequences of this result are also given.

1. Introduction

Let be the class of functions of the following form: which are analytic in . We denote by the subclass of consisting of univalent functions and by and the usual subclasses of whose members are starlike (with respect to the origin) and close-to-convex, respectively. Finally, we denote by the family of functions which satisfy the condition . It is well known that .

Let be the class of functions of the following form: which are analytic in . If satisfies , then we say that is the Carathéodory function. For Carathéodory functions, Nunokawa et al. [1] have shown some sufficient conditions applying the differential inequalities. In the present paper, using the method of differential subordinations, we derive certain conditions under which we have , where . Our results generalize or improve some results due to [1–4].

To prove our results, we need the following lemma due to Miller and Mocanu [5].

Lemma 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 , and(ii).
If is analytic in , with , and then and is the best dominant of (4).

2. Main Results

Theorem 2. Let , , , and . If satisfies and where then and is the best dominant of (5).

Proof. Let , , , and choose Then is analytic and univalent in , , , and satisfy the conditions of the lemma. The function is univalent and starlike in because Further, we have Since , from (11) it is easy to know that Hence the function is close-to-convex and univalent in . Now it follows from (5)–(12) that Therefore, by virtue of the lemma, we conclude that and is the best dominant of (5). The proof of the theorem is complete.

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

Corollary 3. Let , . If satisfies then  .

Proof. Let , , , , , , and in the theorem, then we have satisfy the conditions of the lemma. Note that Hence, similar to the proof of the theorem, we conclude that , that is, .

Remark 4. Note that Corollary 3 was also proved by Nunokawa et al. [1] using another method.
Taking , , , , , and in the theorem, we have the following.

Corollary 5. Let , , satisfies , and then .

Remark 6. Note that the function maps onto the -plane slit along the half-lines , and , . Hence Corollary 5 coincides with the result obtained by Nunokawa et al. [1, Theorem 2] using another method.
Letting , , in Corollary 5, we have the following.

Corollary 7. If , , and where is real and , then .

Remark 8. Lewandowski et al. [3] proved if , , , and then . We see that Corollary 7 improves this result.

Corollary 9. Let with . If for some , then and

Proof. Letting , , then , Taking , and , in the theorem, it follows from (5), (6), and (21) that , which implies that (see [6]).

Remark 10. Frassin and Darus [2] have shown that if and for some , then .
For , the function is analytic and convex univalent in . Since the disk is properly contained in . Therefore, in view of (21), we see that Corollary 9 is better than the result given in [2].
Letting , , , , , , and in the theorem, we have the following.

Corollary 11. If , , , and then .
For the univalent function , we now find the image of the unit disk .
Let , where and are real. We have Elimination of yields Therefore, we conclude that

Remark 12. Nunokawa and Hoshino [4] have proved that if , and then . For , properly contains the half plane . Hence Corollary 11 with is better than the result given in [4].

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant no. 11171045). The author would like to thank the referees for their careful reading and for making some valuable comments which have essentially improved the presentation of this paper.