Inequalities for products of polynomials I

TL;DR

This paper investigates inequalities relating the norms of polynomials and their products, establishing that the minimal constant is for disks and conjecturing the maximal for segments among compact sets.

Contribution

It proves the minimal constant for the inequalities is 2 for disks and conjectures it is largest for segments among all compact connected sets.

Findings

The best constant is 2 for the disk.

Conjecture: the largest constant occurs for segments.

Asymptotically sharp constants are known for arbitrary compact sets.

Abstract

We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the Gelfond-Mahler inequality for the unit disk and the Kneser-Borwein inequality for the segment $[-1,1]$. Furthermore, the asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. It is shown here that this best constant is smallest (namely: 2) for a disk. We also conjecture that it takes its largest value for a segment, among all compact connected sets in the plane.