Functional inequalities for max eigenvalues of nonnegative matrices

Let Pn+ denote the set of all n×nn×n nonnegative matrices. For a function f:R+m→R+ and matrices Ak=(aijk)i,j=1n, k=1,…,mk=1,…,m, definef(A1,…,Am)=(f(aij1,…,aijm))ij=1n.For each A∈Pn+ we denote its spectral radius by ρ(A)ρ(A) and its max eigenvalue by μ(A)μ(A). In a previous paper, all functions ff which satisfyρ(f(A1,…,Am))⩽f(ρ(A1),…,ρ(Am)),∀n∈N,∀A1,…,Am∈Pn+and some functions which satisfyf(ρ(A1),…,ρ(Am))≤ρ(f(A1,…,Am)),∀n∈N,∀A1,…,Am∈Pn+,were characterized. Here, for an interval II in R+R+, we characterize those functions ff satisfyingμ(f(A1,…,Am))⩽f(μ(A1),…,μ(Am)),∀n∈N,∀A1,…,Am∈Innas well as the functions satisfying f(0)=0f(0)=0 andf(μ(A1),…,μ(Am))⩽μ(f(A1,…,Am)),∀n∈N,∀A1,…,Am∈Inn.