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Versions of Koebe 1/4 theorem for analytic and quasiregular harmonic functions and applications. (English) Zbl 1274.30089
Summary: We mainly survey results obtained in [the author, Ann. Acad. Sci. Fenn. 32, No. 2, 301–315 (2007; Zbl 1221.30063)]. For example, we give an elementary proof of two versions of the Koebe $$1/4$$ theorem for analytic functions. 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.

##### MSC:
 30C62 Quasiconformal mappings in the complex plane 30C35 General theory of conformal mappings
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