Inequalities on a Class of Function Sets

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 ${\alpha }_1\le \cdots \le {\alpha }_n$ and ${\sum }_{k=1}^n{\alpha }_k=0$, then $$\sum _{k=1}^n\phi (k{\alpha }_k)\ge 0$$ for every odd function $\phi$ that is increasing and convex on $[0,\mathrm{\infty })$. 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.