Three Value Ranges for Symmetric Self-mappings

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.

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