Two-point distortion theorems for harmonic mappings. (English) Zbl 1207.30035

The two-point distortion of a function \(f(z)\) gives the lower bound of the distance \(|f(z_1) - f(z_2)|\) with respect to the distance between \(z_1\) and \(z_2\). If the distortion is positive, then \(f\) is injective (univalent). Some earlier results gave sufficient conditions for curves in \(\mathbb R^n\) to be injective and for a lift of a harmonic mapping \(f(z) = u(z) + iv(z)\) to a suitable minimal surface to be injective.
Using a real analog of the Schwarzian derivative, the authors improve these results estimating the distortion (which implies injectivity).


30C99 Geometric function theory
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text: arXiv Euclid


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