On Some Functional-differential Inequalities Related to the Exponential Mapping

Abstract

We consider real-valued twice differentiable functions defined on an open interval. Our main result states that if a function $f$ satisfies some inequalities then a map $x\mapsto f(x)\exp(-cx)$ is convex, where $c$ is an arbitrary point of $\mathbf{R}$ or of $\mathbf{R}\setminus (c_1,c_2)$ for some real $c_1, c_2$.