A Gamma envelope and sharp moment inequalities for Gaussian quadratic forms
We study extremal absolute central moments of Gaussian quadratic forms under a fixed Frobenius norm. For a nonzero real symmetric matrix $M$ and $G\sim N(0,I_n)$, we construct an explicit centered difference of Gamma variables with the same mean, variance, and third centered moment as $G^{\mathsf T}MG-\operatorname{tr}M$. After normalization, replacing the quadratic form by this Gamma difference does not decrease $\mathbb{E} f$ for every $C^2$ test function $f$ such that $f''$ is convex and $f$, $f'$, and $f''$ have polynomial growth. In particular, it gives an explicit upper bound for every absolute moment of order $p\ge3$. We then prove that, for every $p\ge4$, this bound is maximized by the centered square $g^2-1$ of a single standard Gaussian variable $g$. The resulting sharp inequality is \[ \bigl\|G^{\mathsf T}MG-\operatorname{tr}M\bigr\|_p \le \|g^2-1\|_p\,\|M\|_{\mathrm F}, \] with equality if and only if $M$ has rank one.