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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 $σ_1^2, σ_2^2$ and the correlation coefficient $ρ$, we prove that for any real numbers $α_1, α_2\in (-1,0)$ or $α_1, α_2\in (0,\infty)$, it holds that %there exist functions of $α_1, α_2$ and $ρ$ such that $${\bf E}[|X_1|^{α_1}|X_2|^{α_2}]-{\bf E}[|X_1|^{α_1}]{\bf E}[|X_2|^{α_2}]\ge f(σ_1,σ_2,α_1, α_2, ρ)\ge 0, $$ where the function $f(σ_1,σ_2,α_1, α_2, ρ)$ will be given explicitly by Gamma function and is positive when $ρ\neq 0$. When $-1<α_1<0$ and $α_2>0,$ Russell and Sun (arXiv: 2205.10231v1) proved the &#34;opposite Gaussian product inequality&#34;, of which we will also give a quantitative version. These quantitative inequalities are derived by employing the hypergeometric functions and the generalized hypergeometric functions.