Holomorphic Mappings of a Strip into Itself with Bounded Distortion at Infinity

A class of holomorphic self-mappings of a strip which is symmetric with respect to the real axis is studied. It is required that the mappings should boundedly deviate from the identity transformation on the real axis. Distortion theorems for this class of functions are obtained, and domains of univalence are found that arise for certain values of the parameter characterizing the deviation of the mappings from the identity transformation on the real axis.