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In Statistics, generalized means of positive random variables are often considered. As is well-known, the generalized mean of order α, {E(v ^α)} ^{1 α} , is smaller (greater) than the ordinary mean if α<1 (α>1). This result can be generalized to a corresponding inequality involving matrix random variables of a specific type. In the special case when α=−1, we have a matrix inequality that has applications in various fields of Statistics. Two such applications are presented.