Distortion theorems for functions regular and bounded in the disk

The paper refines and supplements some classical and modern distortion theorems for functions regular in the disk |z| < 1 and satisfying the condition |f(z)| < 1 for |z| < 1. Distortion bounds at points of the disk |z| < 1 and also on its boundary |z| = 1, provided that |f(z)| = 1, are considered both for univalent and multivalent mappings. Bibliography: 22 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 26–39.