Applications of duality principle to integral transforms of analytic functions

Abstract

Let A be the class of functions analytic in the unit disk and normalized by f(0)=f'(0)+1=0 Let S, S^∗(y), and K(y) be respectively the classes of normalized univalent functions, starlike functions of order y and convex functions of order y. In this paper we investigate the properties of the integral transform V_λ(f) = ∫₀¹ λ(t) f(tz)/t dt, f ∈ A where λ is a non-negative real valued function normalized by ∫₀¹ λ(t)dt=1. From our main results we get conditions on λ and the class Fso that V_λ(f) maps F 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 of S or S^∗(y) or K(y), then the function φ(3,3+ α_z) ^∗ 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.