Cauchy inequalities for the spectral radius of products of diagonal and nonnegative matrices

Abstract:

Inequalities for convex functions on the lattice of partitions of a set partially ordered by refinement lead to multivariate generalizations of inequalities of Cauchy and Rogers-Hölder and to eigenvalue inequalities needed in the theory of population dynamics in Markovian environments: If $A$ is an $n\times n$ nonnegative matrix, $n > 1$, $D$ is an $n\times n$ diagonal matrix with positive diagonal elements, $r(\cdot )$ is the spectral radius of a square matrix, $r(A)>0$, and $x \in [1,\infty )$, then $r^{x-1}(A) r(D^xA) \geq r^x(DA)$. When $A$ is irreducible and $A^T A$ is irreducible and $x>1$, then equality holds if and only if all elements of $D$ are equal. Conversely, when $x>1$ and $r^{x-1}(A)r(D^xA)=r^x(DA)$ if and only if all elements of $D$ are equal, then $A$ is irreducible and $A^T A$ is irreducible.