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Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality: $$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...