Integral Transforms of Functions to Be in a Class of Analytic Functions Using Duality Techniques

Abstract EN

Let Wβ(α,γ), β<1, denote the class of all normalized analytic functions f in the unit disc D={z∈C:|z|<1} such that Re(eiϕ((1-α+2γ)(f/z)+(α-2γ)f'+γzf″-β))>0, z∈D, for some ϕ∈R with α≥0, γ≥0, and β<1.

Let M(ξ), 0≤ξ≤1, denote the Pascu class of ξ-convex functions given by the analytic condition Re((ξz(zf'(z))'+(1-ξ)zf'(z))/ (ξzf'(z)+(1-ξ)f(z)))>0 which unifies the classes of starlike and convex functions.

The aim of this paper is to find conditions on λ(t) so that the integral transform of the form Vλ(f)(z)=∫01λ(t)(f(tz)/t)dt carry functions from Wβ(α,γ) into M(ξ).

As for the applications, for specific values of λ(t), it is found that several known integral operators carry functions from Wβ(α,γ) into M(ξ).

The results for a more generalized operator related to Vλ(f)(z) are also given.