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It is known that $$ Tr[(AB)^n] \leq Tr (A^n B^n)$$ for $A$, $B$ positive semi-definite matrices. I am looking of a generalization of it for a product of $4$ positive semi-definite matrices of the the form, $$Tr [(ABCD)^n] \leq Tr [(AB)^{2n} ] Tr [(DC)^{2n}]$$ Is there any such inequality? Can someone suggest an approach if the above thing can be proved or disproved?

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