A note on the multivariate generalization of a basic simple inequality

We introduce the multivariate analogue of the well known inequality $1+x\leq $\mathrm{e}$^x$,foranabstractnonnegativerealnumber$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.