Analog of the Montel Theorem for Mappings of the Sobolev Class with Finite Distortion

Abstract

We study the classes of mappings with unbounded characteristic of quasiconformality and obtain a result on the normal families of open discrete mappings $f : D → ℂ \backslash \{a, b\}$ from the class $W\{\text{loc}^{1,1}$ with finite distortion that do not take at least two fixed values $a 6 ≠ b$ in $ℂ$ whose maximal dilatation has a majorant of finite mean oscillation at every point. This result is an analog of the well-known Montel theorem for analytic functions and is true, in particular, for the so-called $Q$-mappings.