# 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)
The Bloch constant of bounded harmonic mappings. (English) Zbl 0677.30020
In this work we observe that a harmonic mapping h of the open unit disk $\Delta$ into itself is Lipschitz when regarded as a function of the hyperbolic disk ($\Delta$,$\rho)$ in ${\bbfC}$, endowed with the euclidean distance, and its Lipschitz number $$\beta\sb h=\sup (\vert h(z)-h(w)\vert)/(\rho (z,w))$$ called the Bloch constant of h, does not exceed 4/$\pi$. Writing $h=f+\bar g$, with f and g analytic, we show that $$\beta\sb h=\sup \{(1-\vert z\vert\sp 2)[\vert f'(z)\vert +\vert g'(z)\vert]:\quad z\in \Delta \},$$ generalizing a well-known formula for analytic functions. We then characterize the extremal ones: $\beta\sb h=4/\pi$ if and only if there exists a constant $\zeta$ of modulus 1 and an analytic function g: $\Delta$ $\to \Delta$ such that $Re(\zeta h)=2/\pi \arg ((1+g)/(1-g))$, where g has Lipschitz number 1 as a map from ($\Delta$,$\rho)$ to itself. Furthermore the imaginary part of $\zeta$ h needs not be the constant 0, unless g is an inner function.
Reviewer: F.Colonna

##### MSC:
 30D50 Blaschke products, etc. (MSC2000)
##### Keywords:
harmonic mapping; Bloch function; inner function
Full Text: