On the solutions for Grötzsch annulus extremal problem

Abstract

Using the properties of planar quasiconformal mappings, we obtain the solutions for the Grötzsch extremal problem on the annulus. We also point out the shortcoming used in [3] for solving this extremal problem. Moreover, by the hyperbolic area distortion and the property of domain module, one criterion for the solution of the Grötzsch annulus extremal problem is given under some conditions.