For $alpha,z>0$ with $alpha e1$, motivated by comparison between different kinds of Renyi divergences in quantum information, we consider log-majorization between the matrix functions begin{align*} P_alpha(A,B)&:=B^{1/2}(B^{-1/2}AB^{-1/2})^alpha B^{1/2}, Q_{alpha,z}(A,B)&:=(B^{1-alphaover2z}A^{alphaover z}B^{1-alphaover2z})^z end{align*} of two positive (semi)definite matrices $A,B$. We precisely determine the parameter $alpha,z$ for which $P_alpha(A,B)prec_{log}Q_{alpha,z}(A,B)$ and $Q_{alpha,z}(A,B)prec_{log}P_alpha(A,B)$ holds, respectively.
Download