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 T λ,p [f ](z) = D λ f (z) + pz(D λ f (z)) p + 1 + D λ f (z) − pz(D λ f (z)) p + 1 , z ∈ D, where the operator D λ 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 T λ,p [f ]. An extension of the above operator is also given.