  We establish that \[\sum_{m=1}^\infty \sum_{n=1}^\infty a_m \overline{a_n}
\frac{mn}{(\max(m,n))^3} \leq \frac{4}{3}\sum_{m=1}^\infty |a_m|^2\] holds for
every square-summable sequence of complex numbers $a = (a_1,a_2,\ldots)$ and
that the constant $4/3$ cannot be replaced by any smaller number. Our proof is
rooted in a seminal 1911 paper concerning bilinear forms due to Schur, and we
include for expositional reasons an elaboration on his approach.
