The best constant in a Hilbert-type inequality

We establish that n-ary sumation infinity m=1 n-ary sumation infinity n=1 aman mn (max(m, n))3 <= 4 3 n-ary sumation infinity m=1 |am|2 holds for every square-summable sequence of complex numbers a = (a1, a2, ...) 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.(c) 2023 The Author(s). Published by Elsevier GmbH. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).