Two sufficient conditions for the univalence of analytic functions

In this article we obtain sufficient conditions for the univalence of $n$-symmetric analytic functions in the region $|\zeta|>-1$ and in the disk $|\zeta|<-1$. We examine the question of univalent variation of functions analytic in $|\zeta|<-1$ and mapping $|\zeta|=1$ onto a contour with two zero angles. We give an application of these results to the fundamental converse boundary-value problems.