Abstract

We obtain certain simple sufficiency criteria for a class of -valent alpha convex functions. Many known results appear as special consequences of our work. Some applications of our work to the generalized integral operator are also given.

1. Introduction

Let denote the class of functions of the form which are analytic and multivalent in the open unit disk . We note that and . Also let , and , (, ,  ,), denote the subclasses of consisting of all functions , of the form (1), which are defined, respectively, by For and , the above three classes reduce to the well-known classes of starlike, convex, and alpha convex functions of order , respectively. We also note that , .

Sufficient condition was studied by various authors for different subclasses of analytic and multivalent functions; for some of the related work, see [1–9]. The object of the present paper is to obtain sufficient conditions for the class of multivalent alpha convex functions of order . We also consider some special cases of our results which lead to various interesting corollaries, and relevances of some of these with other known results are also mentioned.

We will assume throughout our discussion, unless otherwise stated, that , , , and .

To obtain our main results, we need the following lemmas due to Mocanu [10].

Lemma 1. If satisfies the condition then

Lemma 2. If satisfies the condition where is the unique root of the equation then

2. Sufficient Conditions for the Class

Theorem 3. If satisfies then .

Proof. Let us set a function by for . Then clearly (9) shows that .
Differentiating (9) logarithmically, we have which gives Thus using (8), we have Hence, using Lemma 1, we have .
From (10), we can write Since , it implies that . Therefore, we get or and this implies that .
By taking and in Theorem 3, we obtain Corollaries 4 and 5 proved by Goyal et al. [6].

Corollary 4. If satisfies for , then .

Corollary 5. If satisfies for , then .

Further if we take and in Corollaries 4 and 5, we get the following result proved by Uyank et al. [9].

Corollary 6. If satisfies for , then .

Corollary 7. If satisfies for , then .

Remark 8. If we put and in Corollaries 4 and 5, we get the result proved by Mocanu [7] and Nunokawa et al. [8], respectively.

Theorem 9. If satisfies where is the unique root of (6), then .

Proof. Let us set for . Then clearly (9) shows that .
Differentiating (9), we have which gives Thus using (8), we have where is the root of (6). Hence, using Lemma 2, we have . Now by the same arguments as in the proof of Theorem 3, we obtain the required result.

Making in Theorem 9, we have the following.

Corollary 10. If satisfies then .

Further if we take in Corollary 10, we get the following result proved by Uyank et al. [9].

Corollary 11. If satisfies where is the unique root of the equation then belongs to the class of starlike functions of order .

Taking in Theorem 9, we have the following.

Corollary 12. If satisfies then .

Remark 13. If we take and in Corollary 12, we get the result proved in [9], and further for , we get the result proved by Mocanu [10].

3. Generalized Integral Operator

For , we consider Clearly , and when , and , then (29) reduces to the well-known Alexander integral operator [11].

Theorem 14. If , , and satisfy then .

Proof. From (29), we get Differentiating (31), logarithmically, we get Then by simple computation, we have where we have used (30). By using Theorem 3 with , , and , we have.
From (32), we can write or which shows that , where .

Theorem 15. If , , and satisfy where is the unique root of (6), then .

Proof. The result follows on similar lines as in the last theorem using Theorem 9 with , , and .

Conflict of Interests

The authors have no conflict of interests.