Koebe domain in the Montel class
UDC
517.54Abstract
Study subclass of the well-known Montel class of regular simple functions, determinated by additional metric condition. Using the module method in form of general theorem of extreme partition of the plane on non-overlapping admissible domains, solve the problem of finding Koebe domain in the pointed class. In terms of moduli domains of extreme partition, which includes extremal configuration, inner and upper boundary of Koebe domain are described. These boundaries are determinated by special equations. For points of inner boundary obtained extreme mapping of the unit disc to the simply connected domain, associated with some quadratic differential. For upper boundary extremal mappings do not exist, but the boundary can not be improved.
Keywords:
conformal mapping, quadratic differential, extremal decomposition, extremal mappingReferences
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