PPW and Chitti type inequalities for the Robin Laplacian operator

TL;DR

This paper establishes new inequalities for the Robin Laplacian eigenvalues, including an upper bound for the eigenvalue ratio and a reverse Holder inequality for the first eigenfunction, advancing spectral geometry understanding.

Contribution

It provides the first known upper bound for the eigenvalue ratio in Robin problems and generalizes Chiti's reverse Holder inequality to Robin eigenfunctions.

Findings

Proved an upper bound for the ratio of the first two Robin eigenvalues.

Established a reverse Holder inequality for the first Robin eigenfunction.

Connected the results to the PPW conjecture for Robin eigenvalues.

Abstract

In this paper, we prove two results for the Robin eigenvalue problem. One is an upper bound for the ratio of the first two eigenvalues which can be used to recover the PPW conjecture proved by M.S.Ashbaugh and R.D.Benguria, the other is a reverse Holder inequality for the first eigenfunction which is a natural generalization of Chiti's reverse Holder inequality for the first eigenfunction of Dirichlet Laplacian.