Some monotonicity properties of Schur powers of matrices and related inequalities (Q1064381)

Having as a motivation certain probability models, the authors study some inequalities for the spectral radius \(\rho\) of the Schur product of matrices in some conditions. They prove for example: 1. \(\rho\) (A\(\circ B)\leq [\rho (A\circ A)\rho (B\circ B)]^{1/2}\), A and B nonnegative [conditions are given for the equality to occur when A and B are irreducible and when they are stochastic]; 2. If \(\rho_ r=\rho (P^{[r]})\), P nonnegative irreducible, then \(\rho^ 2_ r\leq \rho_{r-1}\rho_{r+1}\) (log \(\rho\) \({}_ r\) convex) for all positive r; 3. For a finite stationary Markov chain, the entropy is given by \((d\rho_ r/dr)_{r=1}\); 4. Inequalities of the form \(\rho (Q^{[\phi]})\leq \rho (P^{[\phi]})\) hold for stochastic matrices and for functions \(\phi\) under given conditions.