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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

30D50Blaschke products, etc. (MSC2000)
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