  We prove the 3-dimensional Gaussian product inequality, i.e., for any
real-valued centered Gaussian random vector $(X,Y,Z)$ and $m\in \mathbb{N}$, it
holds that
${\mathbf{E}}[X^{2m}Y^{2m}Z^{2m}]\geq{\mathbf{E}}[X^{2m}]{\mathbf{E}}[Y^{2m}]{\mathbf{E}}[Z^{2m}]$.
Our proof is based on some improved inequalities on multi-term products
involving 2-dimensional Gaussian random vectors. The improved inequalities are
derived using the Gaussian hypergeometric functions and have independent
interest. As by-products, several new combinatorial identities and inequalities
are obtained.
