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 much attention. In this note, we investigate the quantitative versions of the two-dimensional Gaussian product inequalities. For any centered, nondegenerate, and two-dimensional Gaussian random vector (X_1, X_2) with E[X_1^2]=E[X_2^2]=1 and the correlation coefficient ρ , we prove that for any real numbers α _1, α _2∈ (-1,0) or α _1, α _2∈ (0,∞ ) , it holds that 𝐄[ | X_1| ^α _1| X_2| ^α _2]-𝐄[ | X_1| ^α _1]𝐄[ | X_2| ^α _2]≥ f(α _1, α _2, ρ )≥ 0, where the function f(α _1, α _2, ρ ) will be given explicitly by the Gamma function and is positive when ρ≠ 0 . When -1<α _1<0 and α _2>0 , Russell and Sun (Statist. Probab. Lett. 191:109656, 2022 ) 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.