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arXiv · 1501.04194

Radii of covering disks for locally univalent harmonic mappings

Sergey Yu. Graf·Saminathan Ponnusamy·Victor V. Starkov
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Abstract

For a univalent smooth mapping $f$ of the unit disk $\ID$ of complex plane onto the manifold $f(\ID)$, let $d_f(z_0)$ be the radius of the largest univalent disk on the manifold $f(\ID)$ centered at $f(z_0)$ ($|z_0|<1$). The main aim of the present article is to investigate how the radius $d_h(z_0)$ varies when the analytic function $h$ is replaced by a sense-preserving harmonic function $f=h+\overline{g}$. The main result includes sharp upper and lower bounds for the quotient $d_f(z_0)/d_h(z_0)$, especially, for a family of locally univalent $Q$-quasiconformal harmonic mappings $f=h+\overline{g}$ on $|z|<1$. In addition, estimate on the radius of the disk of convexity of functions belonging to certain linear invariant families of locally univalent $Q$-quasiconformal harmonic mappings of order $α$ is obtained.

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Sergey Yu. Graf, Saminathan Ponnusamy, Victor V. Starkov. 2016-03-23. Radii of covering disks for locally univalent harmonic mappings. https://arxiv.org/abs/1501.04194

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