On the value region of $f(z_0)$ in one class of typically real functions

Let $T_R$ be the class of functions $f(z)$ with $f(0)=0$ and $f'(0)=1$ that are regular and typically real in the disk $|z|<1$. The region of values of the system $\{f(z_0), f(r), f''(0)/2\}$ (for fixed $z_0$ and $r$, $0<r<1$) on the class $T_R$ is determined. The region of values of $f(z_0)$ on the class of functions from $T_R$ with fixed $f(r)$ and $f''(0)$ is found.