On the Hilbert inequality (Q1969403)

Summary: It is shown that the Hilbert inequality for double series can be improved by introducing the positive real number \({1\over \pi^2}\left({s^2(a)\over\|a\|^2}+ {s^2(b)\over\|b\|^2}\right)\), where \(s(x)= \sum^\infty_{n=1} {x_n\over n}\) and \(\|x\|^2= \sum^\infty_{n=1} x^2_n\) \((x= a,b)\). The coefficient \(\pi\) of the classical Hilbert inequality is proved not to be the best possible if \(\|a\|\) or \(\|b\|\) is finite. A similar result for the Hilbert integral inequality is also proved.