Bounded holomorphic functions in a circular annulus

Using 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, we consider differential inequalities for functions that are weakly univalent in the disk. Unsolved problems are given.