Applications of duality principle to integral transforms of analytic functions

Abstract

LetAbe the class of functions analytic in the unit disk and normalized by f(0)=f′(0)+1=0 Let S, S*(γ), and K(γ)be respectively the classes of normalized univalent functions, starlike functions of order γ and convex functions of order γ. In this paper we investigate the properties of the integral transform where λ is a non-negative real valued function normalized by ∫1 0 λ(t)dt=1. From our main results we get conditions on λ and the classFso that V λ (f) mapsF into various subclasses of the class of univalent functions. As a corollary to our results, we give an affirmative answer in support of a conjecture of Kim: f is a member ofSor S*(γ) or K(γ), then the function φ(3,3+ αz) * f(z) belongs to the same class for α >1, where φ(b,c;z) * f(z) stands for the convolution of incomplete beta function with f∈ A