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Hilbert’s and Related Inequalities

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Inequalities Involving Functions and Their Integrals and Derivatives

Part of the book series: Mathematics and Its Applications ((MAEE,volume 53))

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Abstract

The inequality

$${\sum\limits_{m = 1}^\infty {\sum\limits_{m = 1}^\infty {\frac{{{a_m}{b_n}}}{{m + n}} \leqslant \pi } \left( {\sum\limits_{m = 1}^\infty {a_m^2} } \right)} ^{1/2}}{\left( {\sum\limits_{n = 1}^\infty {b_n^2} } \right)^{1/2}}$$
(1.1)

is known as Hilbert’s double series theorem, or Hilbert’s inequality. It was first proved by Hilbert in his lectures on integral equations, and a proof was published by H. Weyl [1] in his dissertation in 1908. The exact value 7r of the constant (which is best possible) was given by I. Schur [2]. He also gave an integral analogue of (1.1).

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Authors and Affiliations

  1. University of Belgrade, Yugoslavia

    D. S. Mitrinović

  2. University of Zagreb, Yugoslavia

    J. E. Pečarić

  3. Iowa State University, Ames, USA

    A. M. Fink

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  1. D. S. Mitrinović
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  2. J. E. Pečarić
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  3. A. M. Fink
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Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1991). Hilbert’s and Related Inequalities. In: Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications, vol 53. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3562-7_5

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  • DOI: https://doi.org/10.1007/978-94-011-3562-7_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5578-9

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