Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-11T03:26:51.679Z Has data issue: false hasContentIssue false
  • English
  • Français

A further generalization of Hilbert's inequality

Part of: Inequalities

Published online by Cambridge University Press:  26 February 2010

Hugh L. Montgomery  and
Jeffrey D. Vaaler
Hugh L. Montgomery
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, U.S.A.
Jeffrey D. Vaaler
Affiliation:
Department of Mathematics, The University of Texas, Austin, TX 78712, U.S.A.
Get access

Extract

Hilbert's inequality asserts that

for arbitrary complex numbers ar. The constant π was first obtained by Schur [5], and is best possible. Following a suggestion of Selberg, Montgomery and Vaughan [4] showed that

where the γr are distinct real numbers and

MSC classification

Secondary: 26D20: Other analytical inequalities
Type
Research Article
Information
Mathematika , Volume 46 , Issue 1 , June 1999 , pp. 35 - 39
Copyright
Copyright © University College London 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Graham, S. W. and Vaaler, J. D.A class of extremal functions for the Fourier Transform Trans. Amer. Math. Soc., 265 (1981), 283–302.CrossRefGoogle Scholar
2.Hardy, G. H., Littlewood, J. E. and Pόlya, G.. Inequalities. (CUP, 1967).Google Scholar
3.Hoffman, K.. Banach Spaces of Analytic Functions. (Prentice-Hall, 1962).Google Scholar
4.Montgomery, H. L. and Vaughan, R. C.. Hilbert's Inequality. J. London Math. Soc. (2) 8 (1974), 73–82.CrossRefGoogle Scholar
5.Schur, I.. Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. J. Reine Angew. Math., 140 (1911), 1–28.CrossRefGoogle Scholar
6.Zygmund, A.. Trigonometric Series. Vols.I, II (CUP, 1968).Google Scholar