Zur, Jan

In this doctoral thesis, we consider the zeros of harmonic mappings in the complex plane. Our study was originally motivated by the theory of gravitational lensing in astrophysics, where special cases of such functions and their zeros play an important role. However, in this work we focus on much more general functions. Our results range from purel...

Sète, Olivier Zur, Jan

We derive a formula for the number of pre-images under a non-degenerate harmonic mapping f, using the argument principle. This formula reveals a connection between the pre-images and the caustics. Our results allow to deduce the number of pre-images under f geometrically for every non-caustic point. We approximately locate the pre-images of points ...

Alkhaleefah, Seraj A. Kayumov, Ilgiz R. Ponnusamy, Saminathan
Published in
Lobachevskii Journal of Mathematics

AbstractIn this paper we first consider another version of the Rogosinski inequality for analytic functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=\sum_{n...

Graf, S. Yu.
Published in
Russian Mathematics

The paper is devoted to finding conditions, sufficient for uniform local univalence of sense-preserving mappings, harmonic in the unit disc of the complex plane; the conditions are given in terms of the generalized Schwarzian derivative introduced by R. Hernández and M. J. Martín. The main section contains proofs of the conditions of univalence and...

Yavuz Duman, Emel Polatoğlu, Yaşar Kahramaner, Yasemin
Published in
Journal of Inequalities and Applications

In the present paper, we give an extension of the idea which was introduced by Sakaguchi (J. Math. Soc. Jpn. 11:72-75, 1959), and we give some applications of this extended idea for the investigation of the class of harmonic mappings. MSC:30C45, 30C55.

Arsenović, Miloš Manojlović, Vesna Vuorinen, Matti
Published in
Journal of Inequalities and Applications

We prove that for harmonic quasiconformal mappings α-Hölder continuity on the boundary implies α-Hölder continuity of the map itself. Our result holds for the class of uniformly perfect bounded domains, in fact we can allow that a portion of the boundary is thin in the sense of capacity. The problem for general bounded domains remains open. MSC 201...

Graf, S. Yu. Eyelangoli, O. R.
Published in
Russian Mathematics

In linear- and affine-invariant families of harmonic mappings of the unit disk we prove some differential inequalities such as the sharp two-sided estimate of the Jacobian and an estimate of the curvature of the image of the circle.

Pingen, Michael
Published in
Analysis

We give a new proof of a well known regularity result for harmonic mappings between Riemannian manifolds due to Giaquinta and Hildebrandt [3]. The proof uses a modification of a method due to L. Caffarelli [2] to show interior and boundary Hölder-continuity of harmonic mappings, whose images lie in a regular ball. In addition a priori estimates are...