  We prove an inequality for the spectral radius of products of non-negative
matrices conjectured by X. Zhan. We show that for all $n\times n$ non-negative
matrices $A$ and $B$, $\rho(A\circ B)\le\rho((A\circ A)(B\circ
B))^{1/2}\le\rho(AB)$, where $\circ$ represents the Hadamard product.
