The best constant in a Hilbert-type inequality

We establish that $$\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{a}_m\overline{a_n}\frac{mn}{(\max (m,n){)}^3}\le \frac{4}{3}\sum _{m=1}^{\mathrm{\infty }}\mathrm{\mid }{a}_m{\mathrm{\mid }}^2$$ holds for every square-summable sequence of complex numbers $a=(a_1,a_2,\dots )$ and that the constant $4\mathrm{/}3$ 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.