By the symmetrization method, new covering and distortion theorems are proved for holomorphic and bounded functions in a circular annulus that preserve one of its boundary components. In particular, inequalities including the Schwarzian derivative at a boundary point of the annulus are established. As corollaries, some differential inequalities for functions that are weakly univalent in the disk are presented. Unsolved problems are formulated.
V. Dubinin (Thu,) studied this question.