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Bloch constants for planar harmonic mappings. (English) Zbl 0956.30012
The authors consider if the well-known theorems, the Landau theorem and the Bloch theorem, from the theory of holomorphic functions are still valid for harmonic mappings, defined in the unit disc $\Cal D$, satisfying similar regularity conditions. It turns out that this is not always possible but one must require some extra assumptions for harmonic mappings. Several theorems are proved: 1)\enspace Two theorems for bounded harmonic mappings similar to the Landau theorem for bounded holomorphic functions are proved. 2)\enspace It is shown by example that there is no Bloch theorem even with normalization $f_z(0)=1$ and $f_{\bar z}(0)=0$. 3)\enspace In order to get a Bloch theorem for harmonic mappings some extra assumption is needed other than this normalization. This extra condition is that the mapping is open. 4)\enspace For $K$-quasiregular harmonic mappings (even in higher dimension), {\it S. Bochner} [Bull. Am. Math. Soc. 52, 715-719 (1946; Zbl 0061.11204)] has already proved the existence of a Bloch constant but has given no estimate. The authors of this paper estimate this Bloch constant in the planar case. 5)\enspace The authors employ their Bloch theorem for quasiregular harmonic mappings to obtain a Bloch theorem for open planar harmonic mappings.

MSC:
30C99Geometric function theory
30C62Quasiconformal mappings in the plane
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