Product inequalities for multivariate Gaussian, gamma, and positively upper orthant dependent distributions

The Gaussian product inequality is an important conjecture concerning the moments of random vectors. While all attempts to prove in full generality have been unsuccessful date, numerous partial results derived recent decades and we provide here further on problem. Most importantly, establish a strong version for multivariate gamma distributions case nonnegative correlations, thereby extending result recently by Genest Ouimet (2021). Further, show that holds with exponents vectors positive components whenever underlying vector positively upper orthant dependent. Finally, negative follows directly from correlation inequality.