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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 837913, 10 pages
On Certain Sufficiency Criteria for -Valent Meromorphic Spiralike Functions
Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan
Received 10 June 2012; Accepted 6 August 2012
Academic Editor: Allan Peterson
Copyright © 2012 Muhammad Arif. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider some subclasses of meromorphic multivalent functions and obtain certain simple sufficiency criteria for the functions belonging to these classes. We also study the mapping properties of these classes under an integral operator.
Let denote the class of functions of the form which are analytic and -valent in the punctured unit disk . Also let and denote the subclasses of consisting of all functions which are defined, respectively, by
We note that for and , the above classes reduce to the well-known subclasses of consisting of meromorphic multivalent functions which are, respectively, starlike and convex of order . For the detail on the subject of meromorphic spiral-like functions and related topics, we refer the work of Liu and Srivastava , Goyal and Prajapat , Raina and Srivastava , Xu and Yang , and Spacek  and Robertson .
Motivated from the work of Frasin , we introduce the following integral operator of multivalent meromorphic functions
For , (1.4) reduces to the integral operator introduced and studied by Mohammed and Darus [12, 13]. Similar integral operators for different classes of analytic, univalent, and multivalent functons in the open unit disk are studied by various authors, see [14–19].
In this paper, first, we find sufficient conditions for the classes and and then study some mapping properties of the integral operator given by (1.4).
We will assume throughout our discussion, unless otherwise stated, that is real with , , , for .
To obtain our main results, we need the following Lemmas.
Lemma 1.1 (see ). If with and satisfies the condition then
Lemma 1.2 (see ). 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 .
2. Some Properties of the Classes and
Theorem 2.1. If satisfies then .
Proof. Let us set a function by
for . Then clearly (2.2) shows that .
Differentiating (2.2) logarithmically, we have which gives
Thus using (2.1), we have
Hence, using Lemma 1.1, we have .
From (2.3), we can write
Since , it implies that . Therefore, we get or and this implies that .
If we take , we obtain the following result.
Corollary 2.2. If satisfies then .
Theorem 2.3. If satisfies then .
Proof. Let us set
Then clearly and . Now
Differentiating logarithmically and then simple computation gives us
Therefore, by using Lemma 1.1, we have which implies that . Since therefore Since , so or
It follows that .
Theorem 2.4. If satisfies
then , where , and
Proof. Let us set
Then is analytic in with .
Taking logarithmic differentiation of (2.22) and then by simple computation, we obtain with
Now for all real and satisfying , we have
Reputing the values of , , , and and then taking real part, we obtain where , , and are given in (2.21).
Let . Then and , for all real and satisfying , . By using Lemma 1.2, we have , that is .
If we put , we obtain the following result.
Corollary 2.5. If satisfies then , where , .
Theorem 2.6. For , let and satisfy (2.9). If then , where .
Proof. From (1.4), we obtain
Differentiating again logarithmically and then by simple computation, we get or, equivalently we can write Now taking real part on both sides, we obtain This further implies that
Let Clearly we have Then by using (2.28) and Corollary 2.2, we obtain Therefore with .
Making use of (2.27) and Corollary 2.5, one can prove the following result.
Theorem 2.7. For , let and satisfy (2.27). If then , where .
The author would like to thank Prof. Dr. Ihsan Ali, Vice Chancellor Abdul Wali Khan University Mardan for providing excellent research facilities and financial support.
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