Radii of covering disks for locally univalent harmonic mappings

Abstract

For a univalent smooth mapping f of the unit disk D of complex plane onto the manifold f (D), let d f (z 0 ) be the radius of the largest univalent disk on the manifold f (D) 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 + 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 + 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.2000 Mathematics Subject Classification. Primary: 30C62, 31A05; Secondary: 30C45,30C75. Key words and phrases. Locally univalent harmonic mappings, linear and affine invariant families, convex and close-to-convex functions, and covering theorems.

The Schwarzian derivatives of harmonic functions and univalence conditions

“…is proved in [13] for the orderα of the affine hull AL of any linear invariant family L. Therefore, |A 2 (ρ, ε)| ≤ α(ρ) + 1. Here α(ρ) denotes order of the harmonic function f ρ (z).…”

Schwarz type lemmas and their applications in Banach spaces

“…If B > 0 is finite, then we call B the Bloch type constant of the set P. One of the long standing open problems of determining the precise value of Bloch type constant of holomorphic mappings with one variable has attracted much attention (see [3,4,20,36,42]). For holomorphic mappings of several complex variables, the Bloch type constant does not exist unless one considers the class of functions under certain constraints.…”

“…In the following, for f = h + g ∈ PH (k), we will use Theorems 2.2 and 2.3 to investigate the ratio B f /B h and give a sharp estimate. For the related studies of the planar harmonic mappings, see [8,10,20].…”