Gasper's determinant theorem, revisited

TL;DR

This paper revisits Gasper's determinant theorem, providing a corrected proof and exploring additional applications of the bounds on the determinant of real matrices based on sums and sums of squares of entries.

Contribution

It offers a corrected proof of Gasper's determinant bound and extends its applications to related matrix inequalities.

Findings

Corrected proof of Gasper's determinant theorem

Extended applications of the determinant bounds

Clarification of conditions for the bounds to hold

Abstract

Let $n \ge 2$ be a natural number, $M$ a real $n \times n$ matrix, $s$ the sum of the entries of $M$ and $q$ the sum of their squares. With $\alpha := s/n$ and $\beta := q/n$, Gasper's determinant bound says that $ |\det M| \le \beta^{n/2}$, and in case of $\alpha^2 \ge \beta$: $$|\det M| \le |\alpha| \left(\frac{n\beta-\alpha^2}{n-1}\right)^{\frac{n-1}2}$$ This article gives a corrected proof of Gasper's theorem and lists some more applications.