The Krzy\.z Conjecture and an Entropy Conjecture

TL;DR

This paper links the Krzyz conjecture to an entropy minimization problem for polynomials with roots on the unit circle, proposing that certain entropy conditions imply the conjecture's validity.

Contribution

It establishes a conditional connection between entropy minimization and the Krzyz conjecture, offering a new approach to its proof under generic hypotheses.

Findings

Entropy minimization for polynomials with roots on the unit circle relates to the Krzyz conjecture.

Conditional proof of the Krzyz conjecture based on entropy and extremum hypotheses.

Provides a new perspective linking polynomial root distributions to complex analysis conjectures.

Abstract

We show that if the minimum entropy for a polynomial with roots on the unit circle is attained by polynomials with equally spaced roots, then, under a generic hypothesis about the nature of the extremum, the Krzyz conjecture on the maximum modulus of the Taylor coefficients of a holomorphic function that maps the disk to the punctured disk is true.