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 < 1g such that f(0) = f0(0) 􀀀 1 = 0 and let P represent the well-known class of Carathéodory functions p such that p(0) = 1 and Re p(z) > 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; & < 1; k  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

Keywords

• Convolution
• Gauss hypergeometric function
• weighted integral transforms

References

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