Convolution Properties of Classes of Analytic and Meromorphic Functions
© Rosihan M. Ali et al. 2010
Received: 30 October 2009
Accepted: 13 May 2010
Published: 13 June 2010
General classes of analytic functions defined by convolution with a fixed analytic function are introduced. Convolution properties of these classes which include the classical classes of starlike, convex, close-to-convex, and quasiconvex analytic functions are investigated. These classes are shown to be closed under convolution with prestarlike functions and the Bernardi-Libera integral operator. Similar results are also obtained for the classes consisting of meromorphic functions in the punctured unit disk.
1. Motivation and Definitions
Let be the set of all analytic functions defined in the unit disk . Denote by the class of normalized analytic functions defined in . For two functions and in , the convolution or Hadamard product of and is the function defined by . A function is subordinate to an analytic function , written , if there exists a Schwarz function , analytic in with and satisfying If the function is univalent in , then is equivalent to and .
which shows that the function is starlike in . Thus, it follows from (1.4) that the component function of is close-to-convex in , and hence univalent. Similarly, the component function of is univalent.
then the class coincides with the class studied in , which there was denoted by , and reduces to a class introduced in  which there was denoted by . It is evident that the classes and extend the classical classes of starlike and convex functions, respectively.
When , the classes and reduce, respectively, to and introduced and investigated in . If , where is defined by (1.8), then the class coincides with studied in . Clearly the classes and extend the classical classes of close-to-convex and quasiconvex functions, respectively.
Theorem 1.3 (see [4, Theorem ]).
In the following section, by using the methods of convex hull and differential subordination, convolution properties of functions belonging to the four classes , , and , are investigated. It would be evident that various earlier works, see, for example, [5–10], are special instances of our work.
In Section 3, new subclasses of meromorphic functions are introduced. These subclasses extend the classical subclasses of meromorphic starlike, convex, close-to-convex, and quasiconvex functions. Convolution properties of these newly defined subclasses will be investigated. Simple consequences of the results obtained will include the work of Bharati and Rajagopal  involving the function , , as well as the work of Al-Oboudi and Al-Zkeri  on the modified Salagean operator.
2. Convolution of Analytic Functions
When , various known results are easily obtained as special cases of Theorem 2.1. For instance, [1, Theorem , page 336] is easily deduced from Theorem 2.1(1), while [1, Corollary , page 336] follows from Theorem 2.1(2). If is defined by (1.8), then [3, Theorem , page 110] follows from Theorem 2.1(1), and [3, Corollary , page 111] follows from Theorem 2.1(2).
The second result is proved in a similar manner.
If is defined by (1.8), then Corollary 2.3 reduces to [2, Theorem , page 324].
Again when , known results are easily obtained as special cases of Theorem 2.5. For instance, [1, Theorem , page 337] follows from Theorem 2.5(1), and [1, Theorem , page 339] is a special case of Theorem 2.5(2).
The proof is similar to the proof of Corollary 2.3 and is therefore omitted.
If is defined by (1.8), then Corollary 2.7 reduces to [2, Theorem , page 326].
3. Convolution of Meromorphic Functions
Here four classes , , , and of meromorphic functions are introduced and the convolution properties of these new subclasses are investigated. As before, it is assumed that is a fixed integer, is a fixed function in , and is a convex univalent function with positive real part in satisfying .
then the class coincides with investigated in .
We shall require the theorem below which is a simple modification of Theorem 1.3.
( ) When , various known results are easily obtained as special cases of Theorem 3.4. For instance, if is defined by (3.7), then [9, Theorem , page 1265] follows from Theorem 3.4(1).
The second result is established analogously.
Again we take note of how our results extend various earlier works. If is defined by (3.7), then [7, Proposition , page 512] follows from Corollary 3.6. If is defined by (3.8), then Corollary 3.6 yields [5, Theorem , page 4]. If is defined by (3.9), then Corollary 3.6 reduces to [6, Theorem , page 11].
The proof is analogous to Corollary 2.3 and is omitted.
The work presented here was supported in part by research grants from Universiti Sains Malaysia and University of Delhi.
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