Absolute moment inequalities under quadratic-form positivity

Abstract

We prove the open question posed by Zhuang and Hu in Remark 3.1. More generally, we consider symmetric joint probability mass functions and joint densities whose associated quadratic form is non-negative. In this class, for every \(r>0\), the inequality \(\E\abs{X+Y}^{r}\ge \E\abs{X-Y}^{r}\) holds for all distributions with finite \(r\)-th absolute moment if and only if \(0<r\le2\).