On a problem in the class of typically real functions

Let $T$ be the class of functions $f(z)=z+\sum^\infty_{n=2}c_nz^n$ regular and typically real in the disk $U=\{z\in\mathbb C\colon|z|<1\}$. In the paper, sharp estimates on the derivative $f'(r)$ ($0<r<1$) for functions in the class $T$ in terms of $f(r)$ and $c_2$ and also $f(r)$, $c_2$, and $c_3$ are obtained.