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

Kalantaris Geometric Modulus Principle describes the local behavior of the modulus of a polynomial. Specifically, if $p(z) = a_0 + sum_{j=k}^n a_jleft(z-z_0right)^j,;a_0a_ka_n eq 0$, then the complex plane near $z = z_0$ comprises $2k$ sectors of angle $frac{pi}{k}$, alternating between arguments of ascent (angles $theta$ where $|p(z_0 + te^{itheta})| > |p(z_0)|$ 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 Kalantaris original paper, we use these extensions to give succinct, elegant new proofs of some classical theorems from analysis.