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 $\alpha_1\leq \cdots\leq \alpha_n$ and $\sum_{k=1}^n \alpha_k=0$, then $$\sum_{k=1}^n \phi(k\alpha_k)\geq 0$$ for every odd function $\phi$ 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.