Three value ranges for symmetric self-mappings of the unit disc

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 $f’(0)\geq 0$ 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.