Convex Combinations of Graham-Kohr Type Extension Operators

Abstract In this paper we discuss about convex combinations of the form K₍, ^, = (1-) ₍, + ₍, K n, λ α, β = (1 - λ) Ψ n, α + λ Ψ n, β of Graham-Kohr type extension operators ₍, Ψ n, α, where 0, 1 λ ∈ 0, 1 and, 0, 1 α, β ∈ 0, 1. The operators ₍, Ψ n, α were defined by I. Graham and G. Kohr in Complex Variables Theory Appl. 47 (2002), 59-72. They proved that the extension operator ₍, Ψ n, α preserve the starlikeness for all n 2 n ≥ 2 and 0, 1 α ∈ 0, 1. The main idea of this paper is to obtain new extension operators defined by convex combinations of Graham-Kohr type operators ₍, Ψ n, α that preserve the starlikeness. We prove that a starlike function f S^* f ∈ S ∗ with Ref' (z₁) >0 Re <mml: