Inequalities on a Class of Function Sets

Abstract

We prove a functional extension of an exponential inequality originally proposed by Bin Zhao and proved by Xiaosheng Mou. The main result asserts that if $\hat{I}{\pm }_1\le \cdots \le \hat{I}{\pm }_n$ and ${\sum }_{k=1}^n\hat{I}{\pm }_k=0$, then \[ \sum_{k=1}^n ϕ(kα_k)\geq 0 \] for every odd function $\ddot{I}•$ that is increasing and convex on $[0,\infty )$. The proof is based on a truncated-sum comparison and the stop-loss characterization of the increasing convex order. As consequences, we recover the original exponential inequality and obtain polynomial and integral variants.