On the regions of values of systems $\{f(z_0),f'(z_0),c_2\}$ and $\{f(r),f'(r),f(z_0)\}$ in the class of typically real functions

Let $T$ be the class of functions $f(z)$ having the following properties: these functions are regular and typically real in the disk $|z|<1$ and have the expansions $f(z)=z+c_2z^2+c_3z^3+\dotsb$. We give algebraic and geometric characterizations of regions of values for the functionals in the class $T$ mentioned in the title. In the same class of functions, we find regions of values for $f'(z_0)$ with fixed $c_2$ and $f(z_0)$ and for $f(z_0)$ with fixed $f(r)$ and $f'(r)$.