Some monotonicity properties of Schur powers of matrices and related inequalities

A series of inequalities are developed relating the spectral radius ?(A ° B) of the Schur product A ° B of two nonnegative matrices A and B with those of ?(A ° A) and ?(B ° B) yielding $\text{?(A ° B) ? [?(A ° A)?(B ° B)]}\text{1}\text{2}$. As a corollary it is proved that the spectral radius of the Schur powers ?_r = ?(A^[r]), A^[r] = A ° A °?°A (r factors) satisfies $\text{(}\text{1}\text{r}\text{)}\text{log}\text{?}{}_{\mathrm{r}}$ is decreasing while $\text{(}\text{1}\text{r?1}\text{)}\text{log}\text{?}{}_{\mathrm{r}}$ is increasing, the latter provided A is a stochastic matrix. The entropy of a finite stationary Markov chain is identified with $\text{d?}{}_{\mathrm{r}}\text{dr|}{}_{\mathrm{r=1}}$. A number of majorization comparisons for the spectral radius of Schur powers is given.