Convolution Properties of Classes of Analytic and Meromorphic Functions

Rosihanm Ali,Kg Subramanian,Moradinargesi Mahnaz,V Ravichandran

doi:10.1155/2010/385728

Rosihanm Ali, Kg Subramanian + Show 2 more

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https://doi.org/10.1155/2010/385728

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Abstract

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.

Highlights

By adding the two inequalities, it is evident that the function f z g z /2 is starlike and both f and g are close-to-convex and univalent
It is evident that the classes Sm∗ g, h and Km g, h extend the classical classes of starlike and convex functions, respectively
Simple consequences of the results obtained will include the work of Bharati and Rajagopal 6 involving the function ka z : 1/ z 1 − z a, a > 0, as well as the work of Al-Oboudi and Al-Zkeri 5 on the modified Salagean operator

Summary

Motivation and Definitions

Let H U be the set of all analytic functions defined in the unit disk U : {z : |z| < 1}. By adding the two inequalities, it is evident that the function f z g z /2 is starlike and both f and g are close-to-convex and univalent This motivates us to consider the following classes of functions. It is evident that the classes Sm∗ g, h and Km g, h extend the classical classes of starlike and convex functions, respectively. By using the methods of convex hull and differential subordination, convolution properties of functions belonging to the four classes Sm∗ g, h , Km g, h , Cm g, h and Qm g, h , are investigated. 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 6 involving the function ka z : 1/ z 1 − z a , a > 0, as well as the work of Al-Oboudi and Al-Zkeri 5 on the modified Salagean operator

Convolution of Analytic Functions

Convolution of Meromorphic Functions