Generalized harmonic Koebe functions
\'Alvaro Ferrada-Salas, Mar\'ia J. Mart\'in
TL;DR
This paper introduces a family of harmonic mappings in the unit disk related to generalized Koebe functions, demonstrating their extremal properties in coefficient growth, distortion, and invariance within harmonic function families.
Contribution
It characterizes harmonic mappings that maximize Taylor coefficients and distortion, extending classical Koebe function properties to harmonic mappings.
Findings
Harmonic mappings maximize Taylor coefficients and growth.
The family relates to classical generalized Koebe functions.
These mappings are extremal in affine and linear invariant families.
Abstract
We present a family of sense-preserving harmonic mappings in the unit disk related to the classical generalized (analytic) Koebe functions. We prove that these are precisely the mappings that maximize simultaneously the real part of every Taylor coefficient as well as the growth and distortion of functions in affine and linear invariant families of complex-valued harmonic functions.
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