Functional inequalities for spectral radii of nonnegative matrices (Q912948)

Let f be a function of nonnegative m-tuples of nonnegative numbers. If \(A_ 1,...,A_ m\) are \(n\times n\) nonnegative matrices, define \(f(A_ 1,...,A_ m)\) to be the \(n\times n\) matrix whose (i,j) element is \(f(a^ l_{ij},...,a^ m_{ij})\) where the superscript is an index not an exponent. Let \(\rho\) (A) denote the spectral radius of A. The paper characterizes all functions which satisfy \(\rho (f(A_ 1,...,A_ m))\leq f(\rho (A_ 1),...,\rho (A_ m)),\) and studies various classes for which \(f(\rho (A_ 1),...,\rho (A_ m))\leq \rho (f(A_ 1,...,A_ m))\).