On the convex sum of certain univalent functions and the identity function

Abstract

If g(z) is analytic, univalent, and convex in the unit disk E = {|z| and lt; 1}, g(0) = 0, g'(0) = 1, we extend some known results about the classification of Hλ(z) = λZ+ (1- λ)g(Z)when λ ≥ 0. In particular, it is proved Hλ is starlike in E for each λ ∈ [0, 1] when g" (0) = 0.