# Inequalities for the fundamental Robin eigenvalue of the Laplacian for box-shaped domains

## TL;DR

This research develops new inequalities for the fundamental Robin eigenvalue of the Laplacian specifically for box-shaped domains, combining theoretical results on functions and their inverses with applications to multidimensional boxes.

## Contribution

The paper introduces novel inequalities for the Robin eigenvalue on box-shaped domains, utilizing new results on positive, decreasing, convex functions and their inverses.

## Key findings

- Established inequalities for Robin eigenvalues on N-dimensional boxes
- Linked function properties to spectral bounds
- Provided theoretical tools for eigenvalue estimation

## Abstract

This document consists of two papers, both submitted, and supplementary material. The submitted papers are here given as Parts I and II. Part I establishes results, used in Part II, 'on functions and inverses, both positive, decreasing and convex'. Part II uses results from Part I to extablish 'inequalities for the fundamental Robin eigenvalue for the Laplacian on N-dimensional boxes'