A note on the multivariate generalization of a basic simple inequality

TL;DR

This paper introduces a multivariate extension of a fundamental inequality, connecting it to ODE solutions, completely monotone functions, and divided differences, expanding the theoretical framework of inequalities.

Contribution

It presents the first multivariate analogue of the inequality 1+x ≤ e^x, derived from ODE analysis and linked to advanced mathematical concepts.

Findings

Established a multivariate inequality analogous to 1+x ≤ e^x

Connected the inequality to the blow-up time of ODE solutions

Linked the result to completely monotone functions and divided differences

Abstract

We introduce the multivariate analogue of the well known inequality $1+x\leq \mathrm{e}^x$, for an abstract non negative real number $x$. The result emerges from the study of the blow up time of certain solutions of the Cauchy problem for a particular ODE. It is also closely related to the notion of completely monotone functions and the theory of divided differences.