Some more counterexamples for Bombieri's conjecture on univalent functions

TL;DR

This paper disproves a specific case of Bombieri's conjecture on univalent functions by analyzing the global minima of a certain real function related to sine functions, expanding the known counterexamples.

Contribution

It provides new counterexamples to Bombieri's conjecture by identifying conditions where the conjecture fails, especially for certain integer pairs and ratios.

Findings

Disproved Bombieri's conjecture in new cases

Identified the minimum of a sine-based function at zero under specific conditions

Extended the set of known counterexamples to the conjecture

Abstract

We disprove a conjecture of Bombieri regarding univalent functions in the unit disk in some previously unknown cases. The key step in the argument is showing that the global minimum of the real function $\big(n\sin{x}-\sin(nx)\big)/\big(m\sin{x}-\sin(mx)\big)$ is attained at $x = 0$ for integers $m>n\geq2$ when $m$ is odd and $n$ is even, $m$ is sufficiently big and $0.5 \leq n/m \leq 0.8194$.