2022/07/20 by Ze-Chun Hu, Han Zhao, Hu, Ze-Chun +3 · 1 citation
Mathematics · #Mathematical Inequalities and Applications
paper · pdf · doi:10.48550/arxiv.2207.09921
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 (X1, X2) with variances σ12, σ22 and the correlation coefficient ρ, we prove that for any real numbers α1, α2∈ (-1,0) or α1, α2∈ (0,∞), it holds that %there exist functions of α1, α2 and ρ such that \bf E[|X1|α1|X2|α2]-\bf E[|X1|α1]\bf E[|X2|α2]≥ f(σ1,σ2,α1, α2, ρ)≥ 0, where the function f(σ1,σ2,α1, α2, ρ) will be given explicitly by Gamma function and is positive when ρ≠ 0. When -1<α1<0 and α2>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.