and

Let Z’, ’ denote the set of all square n x n nonnegative matrices. For A, = (a;j);j_ “ k=l,..., m (k not a power), we set For each A E P,‘, we let p(A) denote its spectral radius. This paper is concerned with the characterization of those functions f: Rf = + R + satisfying either or P(f(A,,...,A,,)) G~(P(A,),.-J~A”,)) (I) j-(p(A,),...,p(A,)) <P(f(A,,...,Am)) (2) for all A,,..., A,,, EP,+ and every n EN. 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 a E int W y, then we totally characterize all f satisfying (2)