The sum of coefficients of bounded univalent functions

We solve the maximal value problem for the functional $\operatorname{Re}\sum_{j=1}^ma_{k_j}$ in the class of functions $f(z)=z+a_2z^2+\dotsb$ that are holomorphic and univalent in the unit disk and satisfy the inequality $|f(z)|<M$. We prove that the Pick functions are extremal for this problem for sufficiently large $M$ whenever the set of indices $k_1,\dots,k_m$ contains an even number.