Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings (original) (raw)

Schwarzian derivative criteria for valence of analytic and harmonic mappings

Mathematical Proceedings of the Cambridge Philosophical Society, 2007

For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass–Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally, certain classes of harmonic mappings are shown to have finite Schwarzian norm.

Schwarzian Derivatives of Analytic and Harmonic Functions

ee.stanford.edu

In this expository article, we discuss the Schwarzian derivative of an analytic function and its applications, with emphasis on criteria for univalence and recent results on valence. Generalizations to harmonic mappings are then described, using a definition of Schwarzian recently proposed by the authors. Here it is often natural to identify the harmonic mapping with its canonical lift to a minimal surface. §1. Historical background.

A converse to the Schwarz lemma for planar harmonic maps

Journal of Mathematical Analysis and Applications, 2021

A sharp version of a recent inequality of Kovalev and Yang on the ratio of the (H 1) * and H 4 norms for certain polynomials is obtained. The inequality is applied to establish a sharp and tractable sufficient condition for the Wirtinger derivatives at the origin for harmonic self-maps of the unit disc which fix the origin.

Harmonic mappings with analytic part convex in one direction

The Journal of Analysis, 2020

In this paper, we study a family of sense-preserving harmonic mappings whose analytic part is convex in one direction. We first establish the bounds on the pre-Schwarzian norm. Next, we obtain radius of fully starlike and radius of fully convex for this family of harmonic mappings.