Fekete-Szegö problem for close-to-convex functions with respect to a certain convex function dependent on a real parameter

Abstract

Given α ∈[0, 1], let h_α(z) := z/(1 − αz), z ∈ \mathbb{D} := {z ∈ \mathbb{C}: |z| <1}. An analytic standardly normalized function f in \mathbb{D} is called close-to-convex with respect to hα if there exists δ ∈ (−π/2, π/2) such that Re{e^iδzf′(z)/h_α(z)} >0, z ∈ \mathbb{D}. For the class C(h_α) of all close-to-convex functions with respect to h_α, the Fekete-Szegö problem is studied.

Key words： Fekete-Szegö problem; close-to-convex functions; close-to-convex functions with argumentδ; close-to-convex functions with respect to a convex function; functions of bounded turning

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