We prove that any random variable $X$ whose moment generating function is point-wise upper bounded by that of $ G \sim \mathcal{N}(0,1) $ must be dominated by $ G/\mathbb{E}[|G|] $ in convex order, meaning $ \mathbb{E}[f(X)] \le \mathbb{E}[f(G/\mathbb{E}[|G|])] $ for all convex $f$. Equality is attained by taking $ X \sim \mathrm{Unif}(\{-1,1\}) $ and $ f(x) = |x| $.

翻译：我们证明，任何矩生成函数逐点受 $G \sim \mathcal{N}(0,1)$ 矩生成函数上界的随机变量 $X$，在凸序意义下必被 $G/\mathbb{E}[|G|]$ 所控制，即对所有凸函数 $f$ 有 $\mathbb{E}[f(X)] \le \mathbb{E}[f(G/\mathbb{E}[|G|])]$。当取 $X \sim \mathrm{Unif}(\{-1,1\})$ 且 $f(x) = |x|$ 时等式成立。