Abstract
The aim of the present paper is to obtain sufficient condition for the class of meromorphic -valent alpha convex functions of order and then to study mapping properties of the newly defined integral operators. Many known results appeared as special consequences of our work.
1. Introduction
Let denote the class of meromorphic functions normalized by which are analytic and -valent in the punctured unit disk . In particular, , , and . For which is real with , , , and , , we denote by , , and , the subclasses of consisting of all meromorphic -valent functions of the form (1) which are defined, respectively, by Making , in (2), we get the well-known subclasses of consisting of meromorphic-valent functions which are starlike, convex, and alpha convex of order , respectively. For details of the classes defined by (2) and related topics, we refer the raeder to the work of Aouf and Hossen [1], Aouf and Srivastava [2], Ali and Ravichandran [3], Goyal and Prajapat [4], Joshi and Srivastava [5], Liu and Srivastava [6], Raina and Srivastava [7], Xu and Yang [8], and Owa et al. [9].
For , Wang et al. [10] and Nehari and Netanyahu [11] introduced and studied the subclass of consisting of functions satisfying We now extended this concept to define a subclass of consisting of functions of the form (1) satisfying For and in (4), we obtain the classes and of , respectively, studied by Arif [12]; also see [13, 14].
Integral operators for different classes of analytic, univalent, and multivalent functions in the open unit disk are studied by various authors; see [15–21]. We now define the following general integral operator of meromorphic-valent functions: For , we obtain the integral operator studied recently in [22, 23], and, further for , we obtain the integral operator introduced and studied by Mohammed and Darus [24].
Sufficient conditions were studied by various authors for different subclasses of analytic and multivalent functions; for some of the related work see [25–27]. The object of the present paper is to obtain sufficient conditions for the class and then study mapping properties of the integral operator given by (5). We also consider some special cases of our results which lead to various interesting corollaries and relevance of some of these results with other known results which are also mentioned.
We will assume throughout our discussion, unless otherwise stated, that is real with , , ,, , for , , , and
To obtain our main results, we need the following lemmas.
Lemma 1 (see [27]). If with and satisfies the condition then
Lemma 2 (see [28]). Let satisfy the following condition:
If the function is analytic in and
then
2. Sufficiency Criteria for the Class
In this section we establish a new sufficiency criteria for the subclass of .
Theorem 3. If satisfies then , where is given by (6).
Proof. Let us set a function by
for . Then clearly (13) shows that .
Logarithmic differentiating of (13) gives
which further implies
Thus using (12), we get
Therefore by Lemma 1, we have .
From (14), we can write
Since , it implies that . Therefore, we get
or
And therefore .
By taking and in Theorem 3, we obtain Corollaries 4 and 5, respectively, proved by Arif [12].
Corollary 4. If satisfies then .
Corollary 5. If satisfies then .
Remarks. We note that by simple computation (13) gives By taking suitable meromorphic starlike function for in (22) such as , which satisfies the inequality of Lemma 1, we can conclude that the functionof (22) is the subclass of a meromorphic function.
3. Some Properties of the Integral Operator
In this section, we discuss some mapping properties of the integral operator .
Theorem 6. For , let and satisfy (20). If then with , and is given by (5).
Proof. From (5), we obtain
Dividing both sides by , we have
Differentiating again logarithmically, we have
Now by simple computation, we get
or equivalently we have
By taking real part on both sides, we obtain
which further implies that
Let
Clearly we have
Then by using (23) and Theorem 3 with , we obtain
Therefore with .
Making in Theorem 6, we have the following.
Corollary 7. For , let and satisfy (20). If then with .
Theorem 8. For , let . If then , where with .
Proof. From (26), we obtain Let such that is analytic in with . Then (36) can be written as Taking real part on both sides, we have where we have used (35) and the assumption that . Let us put Then, for such that , we have Thus using Lemma 2, we conclude that , which is equivalent to That is, .
Acknowledgment
The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02.