A bound for the coefficient $c_4$ for one-sheeted functions in terms of $|c_2|$

In the class $S$ of functions $f(z)=z+\sum_{k=2}^\infty c_kz^k$ which are regular and single-sheeted in the circle $|z|<1$, the bound for $|c_4|$ in terms of $|c_2|$, obtained by Al'fors, is improved. The crudest bound $|c_4|\leqslant4/15(11+|c_2|)$ is better than that of Al'fors: $|c_4|\leqslant(4/\sqrt{15})\sqrt{11+|c_2|^2}$.