  We prove the three-dimensional Gaussian product inequality (GPI)
$E[X_1^{2}X_2^{2m_2}X_3^{2m_3}]\ge E[X_1^{2}]E[X_2^{2m_2}]E[X_3^{2m_3}]$ for
any centered Gaussian random vector $(X_1,X_2,X_3)$ and $m_2,m_3\in\mathbb{N}$.
We discover a novel inequality for the moment ratio $\frac{|E[
X_2^{2m_2+1}X_3^{2m_3+1}]|}{E[ X_2^{2m_2}X_3^{2m_3}]}$, which implies the
3D-GPI. The interplay between computing and hard analysis plays a crucial role
in the proofs.
