Functional inequalities for spectral radii of nonnegative matrices

AbstractLet P+n denote the set of all square n × n nonnegative matrices. For Ak= (akij)ni,j=1, k=1,…, m (k not a power), we set f(A1,…, Am) = (f(a1ij,…, amij))ni,j = 1. For each AϵP+n, we let ρ(A) denote its spectral radius. This paper is concerned with the characterization of those functions f:Rm+→R+ satisfying either ρ(f(A1,…, Am))⩽f(ρ(A1),…, ρ(Am)) (1) or f(ρ(A1),…, ρ(Am))⩽ρ(f(A1,…, Am)) (2) for all A1,…, AmϵP+n and every n ε N. We totally characterize all functions satisfying (1). We delineate various classes of functions which satisfy (2). If f(0)=0, and f is bounded above in some neighborhood of any αεintRm+, then we totally characterize all f satisfying (2)