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On the extended Hilbert’s inequality. (English) Zbl 0935.26011

Hilbert’s inequality is well known. See, for example, G. H. Hardy, J. E. Littlewood and G. Pólya [“Inequalities” (1934; Zbl 0010.10703; 2nd ed. 1952; Zbl 0047.05302; reprint of the 2nd ed. 1988; Zbl 0634.26008)]. It has been extended from \(\ell_2\) norm to \(\ell_p\) norm. Also the corresponding inequality is sharp. The authors manage to bring the sharp constant inside the \(\ell_p\) and \(\ell_q\) norms where \(p>1\) and \(1/p+1/q=1\) and give a new version of Hilbert’s inequality involving sequences. See also M. Gao [J. Math. Anal. Appl. 212, No. 1, 316-323 (1997; Zbl 0890.26011)].

MSC:

26D15 Inequalities for sums, series and integrals
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References:

[1] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge Univ. Press, 1952.
[2] D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. · Zbl 0199.38101
[3] Ming Zhe Gao, An improvement of the Hardy-Riesz extension of the Hilbert inequality, J. Math. Res. Exposition 14 (1994), no. 2, 255 – 259 (Chinese, with English and Chinese summaries). · Zbl 0816.26005
[4] Zhao Dejun, On a Refinement of Hilbert’s Double Series Theorem, Math. In Practice and Theory, Beijin, China. · Zbl 1135.42306
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