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.

中文翻译：

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三维高斯积不等式

我们证明了 3 维高斯积不等式，即对于任何实值中心高斯随机向量 $(X,Y,Z)$ 和 $m\in \mathbb{N}$，它认为 ${\mathbf{ E}}[X^{2m}Y^{2m}Z^{2m}]\geq{\mathbf{E}}[X^{2m}]{\mathbf{E}}[Y^{2m}] {\mathbf{E}}[Z^{2m}]$。我们的证明基于涉及二维高斯随机向量的多项乘积的一些改进的不等式。改进的不等式是使用高斯超几何函数导出的，并且具有独立的兴趣。作为副产品，获得了几个新的组合恒等式和不等式。