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Subclasses of Meromorphic Functions Associated with Convolution
Journal of Inequalities and Applications volume 2009, Article number: 190291 (2009)
Abstract
Several subclasses of meromorphic functions in the unit disk are introduced by means of convolution with a given fixed meromorphic function. Subjecting each convolutedderived function in the class to be subordinated to a given normalized convex function with positive real part, these subclasses extend the classical subclasses of meromorphic starlikeness, convexity, closetoconvexity, and quasiconvexity. Class relations, as well as inclusion and convolution properties of these subclasses, are investigated.
1. Introduction
Let be the set of all analytic functions defined in the unit disk . We denote by the class of normalized analytic functions defined in . For two functions and analytic in , the function is subordinate to , written as
if there exists a Schwarz function , analytic in with and such that In particular, if the function is univalent in , then is equivalent to and
A function is starlike if is subordinate to and convex if is subordinate to . Ma and Minda [1] gave a unified presentation of these classes and introduced the classes
where is an analytic function with positive real part, , and maps the unit disk onto a region starlike with respect to 1.
The convolution or the Hadamard product of two analytic functions and is given by
In term of convolution, a function is starlike if is starlike, and convex if is starlike. These ideas led to the study of the class of all functions such that is starlike for some fixed function in In this direction, Shanmugam [2] introduced and investigated various subclasses of analytic functions by using the convex hull method [3–5] and the method of differential subordination. Ravichandran [6] introduced certain classes of analytic functions with respect to ply symmetric points, conjugate points, and symmetric conjugate points, and also discussed their convolution properties. Some other related studies were also made in [7–9], and more recently by Shamani et al. [10].
Let denote the class of meromorphic functions of the form
which are analytic and univalent in the punctured unit disk . For we recall that the classes of meromorphic starlike, meromorphic convex, meromorphic closetoconvex, meromorphic convex (Mocanu sense), and meromorphic quasiconvex functions of order , denoted by , , , and respectively, are defined by
The convolution of two meromorphic functions and , where is given by (1.4) and , is given by
Motivated by the investigation of Shanmugam [2], Ravichandran [6], and Ali et al. [7, 11], several subclasses of meromorphic functions defined by means of convolution with a given fixed meromorphic function are introduced in Section 2. These new subclasses extend the classical classes of meromorphic starlike, convex, closetoconvex, convex, and quasiconvex functions given in (1.5). Section 3 is devoted to the investigation of the class relations as well as inclusion and convolution properties of these newly defined classes.
We will need the following definition and results to prove our main results.
Let denote the class of starlike functions of order . The class of prestarlike functions of order is defined by
for , and
Theorem 1.1 (see [12, Theorem 2.4]).
Let , , and . Then, for any analytic function ,
,
where denotes the closed convex hull of .
Theorem 1.2 (see [13]).
Let be convex in and with . If is analytic in with , then
2. Definitions
In this section, various subclasses of are defined by means of convolution and subordination. Let be a fixed function in , and let be a convex univalent function with positive real part in and .
Definition 2.1.
The class consists of functions satisfying in and the subordination
Remark 2.2.
If , then coincides with , where
Definition 2.3.
The class consists of functions satisfying in and the subordination
Definition 2.4.
The class consists of functions such that in for some and satisfying the subordination
Definition 2.5.
For real, the class consists of functions satisfying , in and the subordination
Definition 2.6.
The class consists of functions such that in for some and satisfying the subordination
3. Main Results
This section is devoted to the investigation of class relations as well as inclusion and convolution properties of the new subclasses given in Section 2.
Theorem 3.1.
Let be a convex univalent function satisfying , and with . If , then . Equivalently, if , then .
Proof.
Define the function by
For , it follows that
and therefore,
Hence . A computation shows that
Theorem 1.1 yields
and because , it follows that
Theorem 3.2.
The function if and only if
Proof.
The results follow from the equivalence relations
Theorem 3.3.
Let be a convex univalent function satisfying and with . If , then .
Proof.
Since , it follows that
and thus
Let
A similar computation as in the proof of Theorem 3.1 yields
Inequality (3.9) shows that . Therefore Theorem 1.1 yields
hence .
Corollary 3.4.
under the conditions of Theorem 3.3 .
Proof.
The proof follows from (3.12).
In particular, when , the following corollary is obtained.
Corollary 3.5.
Let and satisfy the conditions of Theorem 3.3 . If , then .
Theorem 3.6.
Let and satisfy the conditions of Theorem 3.3. If , then . Equivalently .
Proof.
If , it follows from Theorem 3.2 that . Theorem 3.3 shows that Hence .
Theorem 3.7.
Under the conditions of Theorem 3.3, if with respect to , then with respect to .
Proof.
Theorem 3.3 shows that . Since , (3.9) yields .
Let the function be defined by
A similar computation as in the proof of Theorem 3.1 yields
Since and , it follows from Theorem 1.1 that
Thus with respect to .
Corollary 3.8.
under the assumptions of Theorem 3.3 .
Proof.
The subordination (3.15) shows that .
Theorem 3.9.
Let . Then
(i),
(ii) for .
Proof.
Define the function by
and the function by
For , it follows that Note also that

(i)
Since and
(3.19)
Theorem 1.2 yields Hence .

(ii)
Observe that
(3.20)
Furthermore and from (i). Since and is convex, we deduce that . Therefore,
Corollary 3.10.
The class is a subset of the class
Proof.
The proof follows from the definition of the classes by taking .
Theorem 3.11.
The function if and only if .
Proof.
If , then there exists such that
Also,
Since , by Theorem 3.2 , . Hence .
Conversely, if , then
for some . Let be such that . The proof is completed by observing that
Corollary 3.12.
Let and satisfy the conditions of Theorem 3.3 . If , then
Proof.
If , Theorem 3.11 gives Theorem 3.7 next gives Thus, Theorem 3.11 yields
Corollary 3.13.
under the conditions of Theorem 3.3.
Proof.
If , it follows from Corollary 3.12 that The subordination
gives . Therefore
Open Problem
An analytic convex function in the unit disk is necessarily starlike. For the meromorphic case, is it true that ?
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Acknowledgment
The work presented here was supported in part by the USM's Reserach University grant and the FRGS grant
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Haji Mohd, M., Ali, R.M., Keong, L.S. et al. Subclasses of Meromorphic Functions Associated with Convolution. J Inequal Appl 2009, 190291 (2009). https://doi.org/10.1155/2009/190291
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DOI: https://doi.org/10.1155/2009/190291
Keywords
 Analytic Function
 Unit Disk
 Meromorphic Function
 Conjugate Point
 Class Relation