Hilbert inequality

A theorem of D. Hilbert on double series:

(*)

where

and the series on the right-hand side have finite positive sums. The constant is precise, i.e. it cannot be decreased. The validity of (*) with was demonstrated by Hilbert, without the precise constant, in his course on integral equations. Its proof was published by H. Weyl [1]. The precise constant was found by I. Schur [2], while the inequality (*) for arbitrary was first quoted by G.H. Hardy and M. Riesz in 1925. There exist integral analogues and generalizations of (*), for example

where is a non-negative kernel, homogeneous of degree , , , , , and

and the previously obtained special case of this inequality [4] with kernel (the so-called double-parametric Hilbert inequality) and constant . The preciseness of this constant has been proved for . It is also asymptotically precise as for an arbitrary admissible fixed . The problem of the asymptotic behaviour of the constant in (*) for finite sums () has not been solved (1988); it is only known that if , the constant is

References

[1]	H. Weyl, "Singuläre Integralgleichungen mit besonderer Berücksichtigung des Fourierschen Integraltheorems" , Göttingen (1908) (Thesis)
[2]	I. Schur, "Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen" J. Reine Angew. Math. , 140 (1911) pp. 1–28
[3]	G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[4]	F.F. Bonsall, "Inequalities with non-conjugate parameters" Quart. J. Math. Oxford (2) , 2 (1951) pp. 135–150
[5]	V. Levin, "On the two-parameter extension and analogue of Hilbert's inequality" J. London Math. Soc. (1) , 11 (1936) pp. 119–124
[6]	N.G. de Bruijn, H.S. Wilf, "On Hilbert's inequality in dimensions" Bull. Amer. Math. Soc. , 68 (1962) pp. 70–73
[7]	P.L. Walker, "A note on an inequality with non-conjugate parameters" Proc. Edinburgh Math. Soc. , 18 (1973) pp. 293–294