  Relatively recently, K.M.R. Audenaert (2010), R.A. Horn and F. Zhang (2010),
Z. Huang (2011), A.R. Schep (2011), A. Peperko (2012), D. Chen and Y. Zhang
(2015) have proved inequalities on the spectral radius and the operator norm of
Hadamard products and ordinary matrix products of finite and infinite
non-negative matrices that define operators on sequence spaces. In the current
paper we extend and refine several of these results and also prove some
analogues for the numerical radius. Some inequalities seem to be new even in
the case of $n\times n$ non-negative matrices.
