On the mutual change of values of the derivative and third coefficient in a class of regular 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 $|z|<1$. Sharp estimates for the derivative $f'(r)$ $(0<r<1)$ in terms of the value $c_3$ and sharp estimates for the coefficient $c_3$ in terms of $f'(r)$ are obtained.