An inequality for means with applications

TL;DR

This paper presents a simple inequality involving means that leads to significant improvements in various mathematical bounds, including those related to the Riemann zeta-function, matrix determinants, and symmetric group characters.

Contribution

It introduces a basic inequality for means that is leveraged to enhance several deep results in number theory, linear algebra, and group theory.

Findings

Improved lower bounds for the Riemann zeta-function on the critical line

Enhanced bounds on determinants of skew-symmetric matrices with ±1 entries

Stronger limits on the maximal order of irreducible symmetric group characters

Abstract

We show that an almost trivial inequality for the first and second mean of a random variable can be used to give non-trivial improvements on deep results. As applications we improve on results on lower bounds for the Riemann zeta-function on the critical line, the determinant of a skew-symmetric matrix with entries $\pm 1$, and on the maximal order of an irreducible character of the symmetric group.