zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 30C99 Geometric function theory 30C62 Quasiconformal mappings in the plane
Keywords:
Bloch constant; harmonic mappings
Full Text: