Cauchy Inequalities for the Spectral Radius of Products of Diagonal and Nonnegative Matrices

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× n nonnegative matrix, n > 1, D is an n× n diagonal matrix with positive diagonal elements, r(·) is the spectral radius of a square matrix, r(A) > 0, and x ∈ [1,∞), then rx−1(A)r(DxA) ≥ rx(DA). When A is irreducible and ATA is irreducible and x > 1, then equality holds if and only if all elements of D are equal. Conversely, when x > 1 and rx−1(A)r(DxA) = rx(DA) if and only if all elements of D are equal, then A is irreducible and ATA is irreducible.