On the Univalence of Derivatives of Functions which are Univalent in Angular Domains

We consider functions $f$ that are univalent in a plane angular domain of angle $\alpha\pi$, $0<\alpha\le2$. It is proved that there exists a natural number $k$ depending only on $\alpha$ such that the $k$th derivatives $f^{(k)}$ of these functions cannot be univalent in this angle. We find the least of the possible values of for $k$. As a consequence, we obtain an answer to the question posed by Kiryatskii: if $f$ is univalent in the half-plane, then its fourth derivative cannot be univalent in this half-plane.