English

Panigrahi T,El-Ashwah R

doi:10.5897/ajmcsr2016.0672

Abstract

By applications of Cauchy-Schwarz inequalities, several sufficient conditions in terms of hypergeometric inequalities were found such that the linear operator preserves and transforms certain well known subclasses of univalent functions to another classes. Relevant connections of our work with the earlier work is pointed out. Key words: Analytic function, subordination, starlike function, convex function, hypergeometric function, Cauchy-Schwarz inequality.

Highlights

Let denote the class of functions f normalized by f (z) = z an zn (1)n=2 which are analytic in the open unit disk where | | k, the image curve f ( ) is a convex arc (Kanas and Wisniowska, 1999)
For k = 0, the classes k UCV and k ST reduce to the classes of convex and starlike functions studied by Robertson (1936) and Silverman (1975) and for k = 1, the aforementioned classes reduce to the classes of uniformly convex and uniformly starlike functions in U studied by Goodman (1991a; b)
Let ( z) be an analytic function with positive real part in U with (0) = 1, ' (0) > 0, which is starlike with respect to 1 and is symmetric with respect to the real axis

Summary

Introduction

By applications of Cauchy-Schwarz inequalities, several sufficient conditions in terms of hypergeometric inequalities were found such that the linear operator Introduced a class R ( ) of functions satisfying the condition: R ( ) := { f Taking (z) = 1 Az ( 1 B < A 1; z U) in 1 Bz (Equation 3), we observe that a function f if and only if the following condition is satisfied: f ' (z) zf '' (z) 1 ( A B) B( f ' (z) zf '' (z) 1) < 1.

Results

Conclusion