A Proof of The Triangular Ashbaugh-Benguria-Payne-P\'{o}lya-Weinberger Inequality

TL;DR

This paper proves that among all triangles, the equilateral triangle maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues, extending previous results to all triangles.

Contribution

It extends the Triangular Ashbaugh-Benguria-Payne-Pólya-Weinberger inequality to all triangles, not just acute ones, using advanced inequalities and variational bounds.

Findings

Equilateral triangle maximizes the eigenvalue ratio among all triangles.

The proof generalizes previous results from acute to all triangles.

Utilizes inequalities by Siudeja and Freitas with improved bounds.

Abstract

In this paper, we show that for all triangles in the plane, the equilateral triangle maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues. This is an extension of work by Siudeja, who proved the inequality in the case of acute triangles. The proof utilizes inequalities due to Siudeja and Freitas, together with improved variational bounds.