The best constant in a Hilbert-type inequality

TL;DR

This paper determines the optimal constant in a Hilbert-type inequality involving double sums and proves it cannot be improved, using classical bilinear form techniques from Schur's 1911 work.

Contribution

It establishes the exact best constant in a specific Hilbert-type inequality and provides a detailed proof based on classical bilinear form methods.

Findings

The inequality holds with constant 4/3 for all square-summable sequences.

The constant 4/3 is proven to be optimal and cannot be reduced.

The proof revisits and elaborates on Schur's classical approach from 1911.

Abstract

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.