The maximum of some functional for holomorphic and univalent functions with real coefficients

In the paper the maximum of the functional $a^{k}_{2}a^{m}_{3}(a_{3}-\alpha a^{2}_{2})$ in the class $S_{R}$ of functions $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}, a_{n}=\overline{a_{n}}$, holomorphic and univalent in the unit disc is obtained for $\alpha$ real and $k, m$ positive integers.