  This paper concerns the problem of determining the optimal constant in the
Montgomery--Vaughan weighted generalization of Hilbert's inequality. We
consider an approach pursued by previous authors via a parametric family of
inequalities. We obtain upper and lower bounds for the constants in
inequalities in this family. A lower bound indicates that the method in its
current form cannot achieve any value below $3.19497$, so cannot achieve the
conjectured constant $\pi$. The problem of determining the optimal constant
remains open.
