Univalent harmonic functions

Hengartner, W.; Schober, G.

doi:10.1090/S0002-9947-1987-0869396-9

Univalent harmonic functions
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by W. Hengartner and G. Schober

Trans. Amer. Math. Soc. 299 (1987), 1-31

DOI: https://doi.org/10.1090/S0002-9947-1987-0869396-9

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Abstract:

Several families of complex-valued, univalent, harmonic functions are studied from the point of view of geometric function theory. One class consists of mappings of a simply-connected domain onto an infinite horizontal strip with a normalization at the origin. Extreme points and support points are determined, as well as sharp estimates for Fourier coefficients and distortion theorems. Next, mappings in $\left | z \right | > 1$ are considered that leave infinity fixed. Some coefficient estimates, distortion theorems, and covering properties are obtained. For such mappings with real boundary values, many extremal problems are solved explicitly.

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