Radii of covering disks for locally univalent harmonic mappings

Article Type

Research Article

Publication Title

Monatshefte fur Mathematik

Abstract

For a univalent smooth mapping f of the unit disk D of complex plane onto the manifold f(D) , let df(z0) be the radius of the largest univalent disk on the manifold f(D) centered at f(z0) (| z0| < 1). The main aim of the present article is to investigate how the radius dh(z0) varies when the analytic function h is replaced by a sense-preserving harmonic function f= h+ g¯. The main result includes sharp upper and lower bounds for the quotient df(z0) / dh(z0) , especially, for a family of locally univalent Q-quasiconformal harmonic mappings f= h+ 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.

First Page

527

Last Page

548

DOI

10.1007/s00605-016-0904-2

Publication Date

7-1-2016

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