A Gaussian expectation product inequality

Let (X1,…,Xn) be any n-dimensional centered Gaussian random vector, in this note the following expectation product inequality is proved: E∏j=1nfj(Xj)≥∏j=1nEfj(Xj) for functions fj,1≤j≤n, taking the forms fj(x)=∫0∞cos(xu)μj(du), where μj,1≤j≤n, are finite positive measures. The motivation of studying such an inequality comes from the Gaussian correlation conjecture (which was recently proved) and the Gaussian moment product conjecture (which is still unsolved). Several explicit examples of such functions fj are given. The proof is built on characteristic functions of Gaussian random variables.