Quantitative Versions of the Two-dimensional Gaussian Product Inequalities

The Gaussian product inequality (GPI) conjecture is one of the most famous inequalities associated with Gaussian distributions and has attracted a lot of concerns. In this note, we investigate the quantitative versions of the two-dimensional Gaussian product inequalities. For any centered non-degenerate two-dimensional Gaussian random vector $(X_1, X_2)$ with variances $\sigma_1^2, \sigma_2^2$ and the correlation coefficient $\rho$, we prove that for any real numbers $\alpha_1, \alpha_2\in (-1,0)$ or $\alpha_1, \alpha_2\in (0,\infty)$, it holds that %there exist functions of $\alpha_1, \alpha_2$ and $\rho$ such that $${\bf E}[|X_1|^{\alpha_1}|X_2|^{\alpha_2}]-{\bf E}[|X_1|^{\alpha_1}]{\bf E}[|X_2|^{\alpha_2}]\ge f(\sigma_1,\sigma_2,\alpha_1, \alpha_2, \rho)\ge 0,$$ where the function $f(\sigma_1,\sigma_2,\alpha_1, \alpha_2, \rho)$ will be given explicitly by Gamma function and is positive when $\rho\neq 0$. When $-1<\alpha_1<0$ and $\alpha_2>0,$ Russell and Sun (arXiv: 2205.10231v1) proved the "opposite Gaussian product inequality", of which we will also give a quantitative version. These quantitative inequalities are derived by employing the hypergeometric functions and the generalized hypergeometric functions.