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I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all $n\geq3$, the function:
$$h(\mu_{1},\ldots,\mu_{n})=\intop_{-c}^{c}\frac{\phi(t)}{\Phi(t)}\left[\sum_{i=1}^{n}\left(\prod_{j=1}^{n}\Phi(t+\mu_{i}-\mu_{j})\right)\right]dt$$
is being minimized on the diagonal, that is: $A=\left\{ (\mu_{1},\ldots,\mu_{n})\in\mathbb{R}^{n}\mid\mu_{1}=\cdots=\mu_{n}\right\}$.

It should be noted that the function is constant over the diagonal and invariant to permutations, therefore, it would have been sufficient to show that it is schur-convex. With that said, numerical estimates indicate that the schur criterion does not hold.

Running simulations show that this indeed the proper answer, but I fail to prove it. Any help would be greatly appreciated.

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