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 (X1,X2) with E[X12</mml:msubsup>]=E[X22</mml:msubsup>]=1 and the correlation coefficient rho, we prove that for any real numbers alpha 1,alpha 2 is an element of(-1,0) or alpha 1,alpha 2 is an element of (0,infinity), it holds that <disp-formula id="Equa">E[|X1|alpha 1|X2|alpha 2]-E[|X1|alpha 1]E[|X2|alpha 2]>= f(alpha 1,alpha 2,rho)>= 0,<graphic position="anchor" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13660_2022_2906_Article_Equa.gif"></graphic></disp-formula> where the function f(alpha 1,alpha 2,rho) will be given explicitly by the Gamma function and is positive when rho not equal 0. When -1<<mml:msub>alpha 1<0 and <mml:msub>alpha 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.