Abstract

The main purpose of this paper is to determine the conditions of starlikeness for certain class of multivalent analytic functions. Relevant connections of the results presented here with those obtained in earlier works are pointed out.

1. Introduction and Main Result

Let denote the class of functions of the form which are analytic in the open unit disk . For convenience, we set . A function is said to be in the class of -valent starlike functions of order in , if it satisfies the following inequality: For simplicity, we write .

In [1], Chichra introduced the class of analytic functions which satisfy the condition He proved that the members of are univalent in . Later, R. Singh and S. Singh [2] showed that . Recently, Gao and Zhou [3] considered the subclass of which is defined by They derived some mapping properties of this class. Moreover, several authors discussed some related analytic function classes associated with the class (see [47]). By using the method of differential subordination, Yang and Liu [8] generalized the above works and studied the subclass of which satisfies the condition where In [9], Owa et al. introduced a new subclass of which satisfies the inequality where , , and (throughout this paper unless otherwise mentioned) the parameters and are constrained as follows: The extreme points, coefficient inequalities, radius of starlikeness, and inclusion relationship for the class are derived. By setting , it is easy to see that the class reduces to the class . If we set in the class , then it reduces to the class , which was studied earlier by Silverman [10], R. Singh and S. Singh [2, 11], independently.

For some recent investigations on the starlikeness of analytic functions, one can refer to [1220]. In the present paper, we aim at deriving the conditions of starlikeness for the class . The main result is presented below.

Theorem 1. Let . Then(1) for , where is the solution of the following equation: (2) for , where is the solution of the following equation:

2. Preliminary Results

In order to establish our main theorem, we will require the following lemmas.

Lemma 2 (see [9]). A function if and only if f can be expressed as follows: where is the probability measure on .

The proof of the following lemma is much akin to that of Theorem 1 which was obtained by Nunokawa et al. [21] (see also Liu [22] and Yang [23]). We, therefore, choose to omit the analogous details involved.

Lemma 3. If satisfies the inequality then .

Lemma 4 (Jack's Lemma [24]). Let be a nonconstant regular function in . If attains its maximum value on the circle at , then where is a real number.

We now give the lower bounds of the following continuous linear operators: acting on the class , which played crucial role in the proof of our main result.

Lemma 5. If , then, for , one has the following.(1)When , then This inequality is sharp.(2)When , then where . The inequality is sharp.

Proof. By Lemma 2, we know that is the extreme function of the class . Thus, we only need to consider the function defined by (17); it follows that We note that (18) can be written as follows: Thus, we find from (19) that Moreover, we observe that the function is convex in , , and maps real axis to real axis; we have Upon substituting (22) into (20) and expanding the integrand into the power series of and integrating it, we can easily get (15). The sharpness of (15) can be seen from (18).
By similarly applying the method of proof of (15), we also can prove (16) holds true. The sharpness of (16) can be found in (17).

3. Proof of Theorem 1

Proof. Suppose that with . It follows from (7) that By noting that we get Thus, we can easily find from (15), (23), and (25) that We now set Then is analytic in with . It follows from (27) that At the same time, we can claim that . Indeed, if not, there exists a point such that by Lemma 4, we obtain For , by virtue of (28), we split it into two cases to prove the following.(1)When , in view of (16), we get Let . If satisfies the condition we have a contradiction to (26) at ; the smallest satisfies (32) is solution of (9). This implies that if and , we have . Thus, we conclude that .(2)When , by means of (15), we have Let . If satisfies the following inequality, we have a contradiction to (26) at ; the smallest satisfies (34) is solution of (10). This shows that if and , we have . It follows from (27) that Therefore, by Lemma 3, we deduce that The proof of the theorem is thus completed.

Remark 6. If (), we cannot find the number such that since the continuous linear operators and acting on do not exist in sharp upper bounds.

Putting in the first part of Theorem 1, we can get the following result.

Corollary 7. Let . Then for , where is the solution of the following equation:

Remark 8. Corollary 7 corrects some errors of Theorem 4 in [3].

By setting in the first part of Theorem 1, we also can get the following criterion for starlikeness obtained by Silverman [10].

Corollary 9. Consider for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grant nos. 11301008, 11226088, and 11101053, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Natural Science Foundation of Educational Committee of Henan Province under Grant 14B110012 of China. The authors are grateful to the referees for their valuable comments and suggestions which essentially improved the quality of this paper.