Abstract

In the present paper, we consider a subclass of -valent analytic functions and obtain certain simple sufficiency criteria by using three different methods for the functions belonging to this class. Many known results appear as special consequences of our work.

1. Introduction

Let be the class of functions analytic and -valent in the open unit disk and of the form

In particular, , , and . By and , and , we mean the subclasses of which are defined, respectively, by

For , , , the previous two classes defined in (2) reduce to the well-known classes of starlike and convex, respectively.

For functions , of the form (1), we define the convolution (Hadamard product) of and by

Now we define the subclass of by

Sufficient conditions were studied by various authors for different subclasses of analytic and multivalent functions, for some of the related work see [1–8]. The object of the present paper is to obtain sufficient conditions for the subclass of . 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 also being mentioned.

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

2. Preliminary Results

To obtain our main results, we need the following Lemma's.

Lemma 1 (see [9]). If with and satisfies the condition then

Lemma 2 (see [10]). If satisfing the condition where is the unique root of the equation then

Lemma 3 (see [11]). Let be a set in the complex plane , and suppose that is a mapping from to which satisfies for and for all real such that . If is analytic in and for all , then .

3. Main Results

Theorem 4. If satisfies then .

Proof. Let us set a function by for . Then clearly (11) shows that .
Differentiating (11) logarithmically, we have which gives
Thus using (10), we have
Hence, using Lemma 1, we have .
From (12), we can write
Since , it implies that . Therefore, we get and this implies that .
Setting and in Theorem 4, we get the following.

Corollary 5. If satisfies then , the class of starlike functions of complex order .

Putting and in Theorem 4, we have the following.

Corollary 6. If satisfies then , the class of convex functions of complex order .

Remark 7. If we put in Corollaries 5 and 6, we get the results proved by Uyanık et al. [1]. Furthermore, for , we obtain the results studied by Mocanu [2] and Nunokawa et al. [3], respectively. Also if we set with and in Theorem 4, we obtain the results due to Goyal et al. [4].

Theorem 8. If satisfies where is the unique root of (8), then .

Proof. Let be given by (11), which clearly belongs to the class .
Now differentiating (11), we have which gives
Thus using (19), we have where is the root of (8). Hence, using Lemma 2, we have .
From (20), we can write
Since , it implies that . Therefore, we get (16), and hence .
Making , with and , we have the following.

Corollary 9. If satisfies where is the unique root of (8) with , then , the class of -valent starlike functions of order .

Also if we take , with and in Theorem 8, we obtain the following result.

Corollary 10. If satisfies where is the unique root of (8) with , then , the class of -valent convex functions of order .

Remark 11. For putting in Corollary 10 and in Corollary 9, we obtain the results proved by Mocanu [10] and Uyanık et al. [1], respectively.

Theorem 12. If satisfies where and
then .

Proof. Let us set
Then is analytic in with .
Taking logarithmic differentiation of (28) and then by simple computation, we obtain with
Now for all real and satisfying , we have
Reputing the values of , , , and then taking real part, we obtain where , , are given in (27).
Let . Then and , for all real and satisfying , . Using Lemma 3, we have . This implies that and hence .
If we put and in Theorem 12, we obtain the following result proved in [12].

Corollary 13. If satisfies then .

Furthermore, for in Corollary 13, we have the following result proved in [13].

Corollary 14. If satisfies then .

Acknowledgment

The author is thankful to the Prof. Dr. Ihsan Ali, Vice chancellor of Abdul Wali Khan University Mardan, for providing research facilities in AWKUM.