Estimating the second coefficient in the class of typically real functions with two function values prescribed

Let $T_1$ be the class of functions $f(z)=z+c_2z^2+\cdots$ regular and typically real in the disk $|z|<1$ whose values $f(r_1)$ and $f(z_2)$ are fixed, $0<r_1<r_2<1$. Let $T_2$ be the class of functions $f(z)=z+c_2z^2+\cdots$ regular and typically real in the disk $|z|<1$ whose values $f(r_1)$ and $f(z_0)$ are fixed, $0<r_1<1$, $0<|z_0|<1$. Sharp estimates on the coefficient $c_2$ are obtained for the classes $T_1$ and $T_2$.