An extension of the Geometric Modulus Principle to holomorphic and harmonic functions

TL;DR

This paper extends Kalantari's Geometric Modulus Principle from polynomials to holomorphic and harmonic functions, providing new proofs of classical analysis theorems.

Contribution

The paper introduces a generalization of the Geometric Modulus Principle to broader classes of functions, namely holomorphic and harmonic functions, enhancing its applicability.

Findings

Extended the principle to holomorphic functions

Extended the principle to harmonic functions

Provided new proofs of classical theorems

Abstract

Kalantari's Geometric Modulus Principle describes the local behavior of the modulus of a polynomial. Specifically, if $p(z) = a_0 + \sum_{j=k}^n a_j\left(z-z_0\right)^j,\;a_0a_ka_n \neq 0$, then the complex plane near $z = z_0$ comprises $2k$ sectors of angle $\frac{\pi}{k}$, alternating between arguments of ascent (angles $\theta$ where $|p(z_0 + te^{i\theta})| > |p(z_0)|$ for small $t$) and arguments of descent (where the opposite inequality holds). In this paper, we generalize the Geometric Modulus Principle to holomorphic and harmonic functions. As in Kalantari's original paper, we use these extensions to give succinct, elegant new proofs of some classical theorems from analysis.