Sergey Yu. Graf | Searcharxiv

arXiv2016

Univalence criterion for harmonic mappings and $Φ$-like functions

In this paper, we obtain a new characterization for univalent harmonic mappings and obtain a structural formula for the associated function which defines the analytic $Φ$-like functions in the unit disk. The new criterion stated in this article for the injectivity of harmonic mappings implies the well-known results of Kas'yanyuk \cite{Kas59} and Brickman \cite{Brick73} for analytic functions, but with a simpler proof than theirs. A number of consequences of the characterization, and examples are also presented. Further investigation provides a new method to construct univalent harmonic mappings with the help of an improved distortion theorem.

math.CV↗

arXiv2016

Radii of covering disks for locally univalent harmonic mappings

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.

math.CV↗