A distortion theorem for univalent gap series

We prove a distortion theorem for conformal mappings of the unit disk for which $\log f'$ is representable as the Hadamard gap series. This theorem implies in particular that such conformal mapping is “almost” bounded, i.e., for every $\varepsilon>0$, there is a positive constant $C_{\varepsilon}$ such that $|f(z)|\leqslant C_{\varepsilon}(1-|z|)^{-\varepsilon}$.