Weighted integral transforms involving convolution with some subclasses of analytic functions

Abstract

Let A represent the class of analytic functions f de ned in the open unit disk U := fz 2 C : jzj 0; z 2 U: A functions p analytic in U such that p(0) = 1 belongs to the class Pk for k 2; if and only if p (z) = 1 2 Z2 0 1 + ze􀀀i 1 􀀀 ze􀀀i d() (z 2 U) ; where () : 0 2 is a function of bounded variation satis es the conditions R2 0 d() = 2 and R2 0 jd()j k: For some 2 R; & 2 and 0; let R k( ; &) denote the class of functions f 2 A satisfying the condition: ei (1 􀀀 ) f (z) z + f0 (z) 􀀀 & 2 Pk (z 2 U) : For f 2 R k( ; &), we de ne the integral transform=m (f) (z) = R1 0 m(t) f(tz) t dt; where m is a non-negative real-valued weight function with R1 0 m(t)dt = 1. The main objective of this paper is to study conditions for invariance of the integral transforms =m and other relevant properties in connection with functions in the class R k( ; &). Also by allowing parameters to vary, we may encompass a large number of previously known results. Key words and Phrases: Convolution; Gauss hypergeom