Three Value Ranges for Symmetric Self-mappings

Abstract

Let $\mathbb{D}$ be the unit disc and $z_0\in \mathbb{D}.$ We determine the value range $\{f(z_0)|f\in {\mathcal{R}}^{\ge }\}$, where ${\mathcal{R}}^{\ge }$ is the set of holomorphic functions $f:\mathbb{D}\to \mathbb{D}$ with $f(0)=0$ and $f^{\prime }(0)\ge 0$ that have only real coefficients in their power series expansion around $0$, and the smaller set $\{f(z_0)\,|\, f\in \mathcal{R}^\geq, \text{$f$ is typically real}\}.$ Furthermore, we describe a third value range $\{f(z_0)|f\in \mathcal{I}\}$, where $\mathcal{I}$ consists of all univalent self-mappings of the upper half-plane $\mathbb{H}$ with hydrodynamical normalization which are symmetric with respect to the imaginary axis.