On the value region of initial coefficients in one class of typically real functions

Let $T$ be the class of functions satisfying the following conditions: these functions are regular and typically real in the unit disk, they have the form $f(z)=z+c_2z^2+c_3z^3+\dotsc$, and the equality $f(z_1)=w_1$ holds for some fixed $z_1$ and $w_1$ with $\operatorname{Im}z_1\ne0$. We find the set of values of the first two coefficients for functions from this class. Boundary functions for these sets of values are found. Some previous results of the author are supplemented. Boundary functions for the sets of values for the functionals $f'(z_1)$ and $f(z_2)$ in the class $T_1$ are found.