  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.
