Two dimensional Berezin-Li-Yau inequalities with a correction term

TL;DR

This paper enhances the Berezin-Li-Yau inequality in two dimensions by introducing a nearly optimal correction term, refining previous bounds and advancing understanding of spectral inequalities.

Contribution

The authors add a positive correction term to the Berezin-Li-Yau inequality in 2D, improving its accuracy and asymptotic behavior over prior results.

Findings

Introduction of a positive correction term improves the inequality.

Asymptotic behavior of the correction term is nearly optimal.

Advances previous results by Melas on spectral inequalities.

Abstract

We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is almost optimal. This improves a previous result by Melas.