On value regions of a functional system in the class of typically real functions

Let $T_R$ be the class of functions
$$ f(z)=z+\sum^\infty_{n=2}c_nz^n $$
that are regular and typically real in the disk $E=\{z\colon|z|<1\}$. For this class, the region of values of the system $\{f(z_0),f(r)\}$ for $z_0\in E$, $r\in(-1,1)$ is studied. The sets
\begin{align*} D_r=\{w\colon w=f(z_0),\ f\in T_R,\ f(r)=a\}\quad&\text{for}\quad-1\le r\le1,\\ \Delta_r=\{(c_2,c_3)\colon f\in T_R,\ -f(-r)=a\}\quad&\text{for}\quad0<r\le1 \end{align*}
are found, where $(r(1+r)^{-2},r(1-r)^{-2})$ is an arbitrary fixed number. Bibl. 11 titles.