Three Value Ranges for Symmetric Self-mappings

TL;DR

This paper characterizes the value ranges of certain symmetric holomorphic self-maps of the unit disk and upper half-plane, focusing on functions with real coefficients, typically real functions, and symmetric univalent functions with hydrodynamical normalization.

Contribution

It provides explicit descriptions of three distinct value ranges for classes of symmetric holomorphic functions, extending classical results to new symmetric and normalization conditions.

Findings

Explicit value ranges for functions with real coefficients and fixed points.

Characterization of value ranges for typically real functions.

Description of value range for symmetric univalent functions with hydrodynamical normalization.

Abstract

Let $\mathbb D$ be the unit disc and $z_0\in\mathbb D.$ We determine the value range $\{f(z_0)\,|\, f\in \mathcal{R}^\geq\}$, where $\mathcal{R}^\geq$ is the set of holomorphic functions $f:\mathbb D\to\mathbb D$ with $f(0)=0$ and $f'(0)\geq0$ 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.