VERSIONS OF KOEBE 1/4 THEOREM FOR ANALYTIC AND QUASIREGULAR HARMONIC FUNCTIONS AND APPLICATIONS

In this paper we mainly survey results obtained in [MM3]. For example, we give an elementary proof of two versions of Koebe 1/ 4 theoremfor analytic functions (see Theorem 1.2 and Theorem 1.4 below). We also show a version of the Koebe theorem for quasiregular harmonic functions. As an application, we show that holomorphic functions (more generally quasiregular harmonic functions) and their modulus have similar behavior in a certain sense.