Functional inequalities for max eigenvalues of nonnegative matrices

Let P-n(+) denote the set of all n x n nonnegative matrices. For a function f : R-+(m) -> R+ and matrices [GRAPHICS] define [GRAPHICS] For each A epsilon P-n(+) we denote its spectral radius by rho(A) and its max eigenvalue by mu(A). In a previous paper, all functions f which satisfy rho(f(A(1),...,A(m))) equal to or less than f(rho(A(1)),...,rho(A(m))), ____n epsilon N, ___(sic)A(1),...,A(m) epsilon P-n(+) and some functions which satisfy f(rho(A(1)),...,rho(A(m))) equal to or less than rho(f(A(1),...,A(m))), ____n epsilon N, ___(sic)A(1),...,A(m) epsilon P-n(+) were characterized. Here, for an interval I in R+, we characterize those functions f satisfying mu(f(A(1),...,A(m))) equal to or less than f(mu(A(1)),...,mu(A(m))), ____n epsilon N, ___(sic)A(1),...,A(m) epsilon I-nn as well as the functions satisfying f(0) = 0 and f(mu(A(1)),...,mu(A(m))) equal to or less than mu(f(A(1),...,A(m))), ____n epsilon N, ___(sic)A(1),...,A(m) epsilon I-nn (c) 2007 Elsevier Inc. All rights reserved.