Harmonic Schwarzian derivative and methods of approximation of zeros

TL;DR

This paper explores the connection between classical derivatives used in complex analysis and iterative methods for zero approximation, extending these relations to harmonic functions.

Contribution

It introduces a novel link between Schwarzian derivatives and zero-finding methods for harmonic functions, expanding classical theory.

Findings

Established relations between derivatives and zero-finding methods for harmonic functions.

Extended classical formulas to harmonic function context.

Provided insights into approximation techniques for harmonic mappings.

Abstract

We review the relation between the classical formulas of the pre-Schwarzian and Schwarzian derivatives of locally univalent analytic functions and the derivatives of the generating functions of the methods due to Newton and Halley, respectively, for approximating zeros. We extend these relations to the cases when the functions considered are harmonic.