Abstract and Applied Analysis
Volume 2015 (2015), Article ID 973613, 6 pages
http://dx.doi.org/10.1155/2015/973613

Research Article

Convolution Properties for Some Subclasses of Meromorphic Functions of Complex Order

M. K. Aouf,¹ A. O. Mostafa,¹ and H. M. Zayed²

¹Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
²Department of Mathematics, Faculty of Science, Menofi…a University, Shebin El Kom 32511, Egypt

Received 3 June 2015; Accepted 5 July 2015

Academic Editor: Allan Peterson

Copyright © 2015 M. K. Aouf et al. 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.

Abstract.

Making use of the operator for functions of the form , which are analytic in the punctured unit disc and , we introduce two subclasses of meromorphic functions and investigate convolution properties, coefficient estimates, and containment properties for these subclasses.

1. Introduction

Let denote the class of meromorphic functions of the form

which are analytic in the punctured unit disc and . Let be given by

then, the Hadamard product (or convolution) of and is given by

We recall some definitions which will be used in our paper.

Definition 1. For two functions and , analytic in , we say that the function is subordinate to in and written , if there exists a Schwarz function , analytic in with and such that   . Furthermore, if the function is univalent in , then we have the following equivalence (see [1]):

Now, consider Bessel’s function of the first kind of order where is an unrestricted (real or complex) number, defined by (see Watson [2, page 40]) (see also Baricz [3, page 7])

which is a particular solution of the second order linear homogenous Bessel differential equation (see, e.g., Watson [2, page 38]) (see also Baricz [3, page 7])

Also, let us define

The operator is a modification of the operator introduced by Szász and Kupán [4] for analytic functions.

By using the Hadamard product (or convolution), we define the operator as follows:

It is easy to verify from (8) that

Definition 2. For , , and , let be the subclass of consisting of function of the form (1) and satisfying the analytic criterion

Also, let be the subclass of consisting of function of the form (1) and satisfying the analytic criterion

It is easy to verify from (10) and (11) that

We note that

(i) and (see Bulboacă et al. [5]);(ii) and (see Aouf [6]);(iii) and    (see Ravichandran et al. [7, with ]).

Definition 3. For and is an unrestricted (real or complex) number, let

It is easy to show that

The object of the present paper is to investigate some convolution properties, coefficient estimates, and containment properties for the subclasses and .

2. Main Results

Unless otherwise mentioned, we assume throughout this paper that and is an unrestricted (real or complex) number.

Theorem 4. If , then if and only if

where , , and also .

Proof. It is easy to verify that

(i) In view of (10), if and only if (10) holds. Since the function is analytic in , it follows that for or for ; this is equivalent to (15) holdding for . To prove (15) for all , we write (10) by using the principle of subordination as

where is Schwarz function, analytic in with and ; hence,

Using (16), (18) may be written as

Thus, the first part of Theorem 4 was proved.
(ii) Reversely, because assumption (15) holds for , it follows that for . This implies that is analytic in (i.e., it is regular in , with ).
Since it was shown in the first part of the proof that assumption (18) is equivalent to (15), we obtain that

Assume that

Relation (20) means that . Thus, the simply connected domain is included in a connected component of . From this, using the fact that and the univalence of the function , it follows that ; this implies that . Thus, the proof of Theorem 4 is completed.

Remark 5. (i) Putting in Theorem 4, we obtain the result obtained by Bulboacă et al. [5, Theorem 1].
(ii) Putting , , and in Theorem 4, we obtain the result obtained by Ponnusamy [8, Theorem 2.1].
(iii) Putting , , , , and in Theorem 4, we obtain the result obtained by Ravichandran et al. [7, Theorem 1.2 with ].

Theorem 6. If , then if and only if

where , , and also .

Proof. Putting

then

From (12) and using the identity

we obtain the required result from Theorem 4.

Remark 7. (i) Putting in Theorem 6, we obtain the result obtained by Bulboacă et al. [5, Theorem 2].
(ii) Putting , , and in Theorem 4, we obtain the result obtained by Ponnusamy [8, Theorem 2.2].

Theorem 8. If , then if and only if

for all .

Proof. If , from Theorem 4, we have if and only if

where , , and also . Since

it is easy to show that (28) holds for if and only if (26) holds. Also,

we may easily check that (28) is equivalent to (27). This completes the proof of Theorem 8.

Theorem 9. If , then if and only if

for all .

Proof. If , from Theorem 6, we have if and only if

where , , and also Since

it is easy to show that (33) holds for if and only if (31) holds. Also,

for , we may easily check that (33) is equivalent to (32). This completes the proof of Theorem 9.

Unless otherwise mentioned, we assume throughout the remainder of this section that is a real number .

Theorem 10. If satisfies inequalities

then .

Proof. We have

which implies inequality (36). Also,

which implies inequality (37). Thus, the proof of Theorem 10 is completed.

Using similar arguments to those in the proof of Theorem 10, we obtain the following theorem.

Theorem 11. If satisfies inequalities

then .

Now, using the method due to Ahuja [9], we will prove the containment relations for the subclasses and .

Theorem 12. For , we have .

Proof. Since , we see from Theorem 8 that

We can write (41) as

since

By using the property, if and , then ; (42) can be written as

which means that . This completes the proof of Theorem 12.

Using the same arguments as in the proof of Theorem 12, we obtain the following theorem.

Theorem 13. For , we have .

Conflict of Interests

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

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