Typically real logharmonic mappings

Zayid Abdulhadi

doi:10.1155/s0161171202111100

Abstract

We consider logharmonic mappings of the form defined on the unit disk U which are typically real. We obtain representation theorems and distortion theorems. We determine the radius of univalence and starlikeness of these mappings. Moreover, we derive a geometric characterization of such mappings.

Highlights

Let H(U ) be the linear space of all analytic functions defined in the unit disk U = {z = x +iy : |z| < 1} and let B be the set of all functions a ∈ H(U) such that |a(z)| < 1 for all z ∈ U
If f is a univalent logharmonic mapping in U, either 0 ∉ f (U) and log f is univalent and harmonic on U, or, if f (0) = 0, f is of the form f (z) = z|z|2βh(z)g(z) where β > −1/2 and |h(z)g(z)| ≠ 0 for z ∈ U, and where F (ζ) = log f is univalent and harmonic in the half plane {ζ : ζ < 0}. Such mappings play an important role in the theory of nonparametric minimal surfaces having a periodic Gauss map
Let f (z) = z|z|2βh(z)g(z) be a univalent logharmonic mapping defined on the unit disk, and h(0) = g(0) = 1, β > −1/2

Summary

Introduction

Denote by TLh the class of all orientation-preserving typically real logharmonic mappings. The class TLh is a compact convex set with respect to the topology of locally uniform convergence and it contains, in particular, the set T of all analytic typically real functions. A corresponding question for univalent typically real logharmonic functions would be as follows.

Objectives

Results

Conclusion