Geodesic
On the Hilbert Inequality
Acta mathematica Universitatis Comenianae, Tome 77 (2008) no. 2 
Zhou Yu; Gao Mingzhe. On the Hilbert Inequality. Acta mathematica Universitatis Comenianae, Tome 77 (2008) no. 2. http://geodesic.mathdoc.fr/item/AMUC_2008_77_2_a14/
@article{AMUC_2008_77_2_a14,
     author = {Zhou Yu and Gao Mingzhe},
     title = {On the {Hilbert} {Inequality}},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {2008},
     volume = {77},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2008_77_2_a14/}
}
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%A Gao Mingzhe
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Voir la notice de l'article provenant de la source Comenius University

In this paper it is shown that the Hilbert inequality for double series~can be improved by introducing a weight function of the form ( Ö n )/(n + 1) ( (( Ö n) – 1)/(( Ö n)+1) – (ln n)/π) ) , where n Î N . A similar result for the Hilbert integral inequality is also given. As applications, some sharp results of Hardy-Littlewood's theorem and Widder's theorem are obtained.