A note on a matrix inequality for generalized means (Q1881076)

The generalized mean of a random variable \(v\) is \(\mu_\alpha:=\{E(v^\alpha)\}^{\frac1\alpha}\). If \(v\) is not restricted to a single point and \(\mu:=Ev\) is finite, the following well-known inequality holds: \[ \mu_\alpha<\mu<\mu_\beta, \] whenever \(\alpha<1<\beta\). The authors consider a tracial version of the inequality, in which \(v\) is multiplied by a matrix of the form \(ww^t\), with \(w\) a random vector with \(Eww^t=I\). After proving their result, the authors show two applications to the theory of measurement error models.