Maximal polynomial subordination to univalent functions in the unit disk

Abstract

Let Ω⊂C be a simply connected domain, 0∈Ω, and let ℘_n,n∈N, be the set of all polynomials of degree at mostn. By ℘_n(Ω) we denote the subset of polynomials p ∈℘_n withp(0)=0 andp(D)⊂Ω, whereD stands for the unit disk {z: |z|<1}, and by

we denote the “maximal range” of these polynomials. Letf be a conformal mapping fromD onto Ω,f(0)=0. The main theme of this note is to relate Ω_n (or some important aspects of it) to the imagesf _s (D), wheref _s (z):=f[(1−s)z], 0<s<1. For instance we prove the existence of a universal constantc ₀ such that, forn≥2c ₀,