Spectral radius of Hadamard product versus conventional product for non-negative matrices

We prove an inequality for the spectral radius of products of non-negative matrices conjectured by Zhan. We show that for all n x n non-negative matrices A and B, rho(A circle B) <= rho(A circle A)(B circle B))(1/2) <= rho(AB), in which circle represents the Hadamard product. (C) 2009 Elsevier Inc. All rights reserved.