UNIVALENT HARMONIC FUNCTIONS GENERATED BY RUSCHEWEYH DERIVATIVES OF ANALYTIC FUNCTIONS

Abstract

For λ ≥ 0, p > 0 and a normalized univalent function f defined on the unit disk D, we consider the harmonic function defined by Tλ,p[f ](z) = Dλf(z) + pz(Dλf(z))′ p+ 1 + Dλf(z)− pz(Dλf(z))′ p+ 1 , z ∈ D, where the operator Dλ is the familiar λ-Ruscheweyh derivative operator. We find some necessary and sufficient conditions for the univalence, starlikeness and convexity as well as the growth estimate of the function Tλ,p[f ]. An extension of the above operator is also given. 2010 Mathematics Subject Classification: 30C45.