## 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).

### MSC:

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

### Keywords:

harmonic mapping; Schwarzian derivative; univalent function
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### References:

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