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 $f$. This is sharp as witnessed by $ X \sim \mathrm{Unif}(\{-1,1\}) $ and $ f(x) = |x| $.

翻译：我们证明，任何矩母函数逐点受控于$ G \sim \mathcal{N}(0,1) $的随机变量$X$，在凸序意义下必然被$ G/\mathbb{E}[|G|] $所支配，即对所有凸函数$f$成立$ \mathbb{E}[f(X)] \le \mathbb{E}[f(G/\mathbb{E}[|G|])] $。该结果是最优的，如$ X \sim \mathrm{Unif}(\{-1,1\}) $与$ f(x) = |x| $所示。