Advances in Operator Theory

Equivalent conditions of a Hardy-type integral inequality related to the extended Riemann zeta function

Michael Th Rassias and Bicheng Yang

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By the use of techniques of real analysis and weight functions, we obtain two lemmas and build a few equivalent conditions of a Hardy-type integral inequality with a non-homogeneous kernel, related to a parameter where the constant factor is expressed in terms of the extended Riemann zeta function. Meanwhile, a few equivalent conditions for two kinds of Hardy-type integral inequalities with the homogeneous kernel are deduced. We also consider the operator expressions.

Article information

Adv. Oper. Theory, Volume 2, Number 3 (2017), 237-256.

Received: 1 March 2017
Accepted: 2 April 2017
First available in Project Euclid: 4 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 65B10: Summation of series

hardy-type integral inequality weight function equivalent form Riemann zeta function operator


Rassias, Michael Th; Yang, Bicheng. Equivalent conditions of a Hardy-type integral inequality related to the extended Riemann zeta function. Adv. Oper. Theory 2 (2017), no. 3, 237--256. doi:10.22034/aot.1703-1132.

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  • B. Arpad and O. Choonghong, Best constant for certain multilinear integral operator, J. Inequal. Appl. 2006, Art. ID 28582, 12 pp.
  • G. H. Hardy, Note on a theorem of Hilbert concerning series of positive terms, Proc. London Math. Soc. (2) 23 (1925), xlv-xlvi (Records for 24 April, 1924). xlv-xlvi.
  • G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, USA, 1934.
  • Y. Hong, On Hardy–Hilbert integral inequalities with some parameters, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Article 92, 10 pp.
  • Y. Hong, On the structure character of Hilbert's type integral inequality with homogeneous kernal and applications (Chinese), J. Jilin Univ. Sci. 55 (2005), no. 2, 189–194.
  • M. Krnić and J. E. Pečarić, Hilbert's inequalities and their reverses, Publ. Math. Debrecen 67 (2005), no. 3-4, 315–331.
  • J. C. Kuang, Real and functional analysis (Continuation)(second volume), Higher Education Press, Beijing, 2015.
  • J. C. Kuang, Applied inequalities, Shangdong Science and Technology Press, Jinan, China, 2004.
  • Y. J. Li and B. He, On inequalities of Hilbert's type, Bull. Austral. Math. Soc. 76 (2007), no. 1, 1–13.
  • G. V. Milovanovic and M. Th. Rassias, Some properties of a hypergeometric function which appear in an approximation problem, J. Global Optim. 57 (2013), no. 4, 1173–1192.
  • D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities involving functions and their integrals and deivatives, Kluwer Academic, Boston, USA, 1991.
  • M. Th. Rassias and B. C. Yang, A multidimensional half - discrete Hilbert - type inequality and the Riemann zeta function, Appl. Math. Comput. 225 (2013), 263–277.
  • M. Th. Rassias and B. C. Yang, On a multidimensional half - discrete Hilbert - type inequality related to the hyperbolic cotangent function, Appl. Math. Comput. 242 (2014), 800–813.
  • I. Schur, Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen (German), J. Reine Angew. Math. 140 (1911), 1–28.
  • Z. Q. Wang and D. R. Guo, Introduction to special functions, Science Press, Beijing, China, 1979.
  • J. S. Xu, Hardy–Hilbert's Inequalities with two parameters, Adv. Math. (China) 36 (2007), no. 2, 189–202.
  • B. C. Yang, The norm of operator and Hilbert-type inequalities, Science Press, Beijing, China, 2009.
  • B. C. Yang, On Hilbert-type integral inequalities and their operator expressions, J. Guangaong Univ. Edu. 33 (2013), no. 5, 1–17.
  • B. C. Yang, Hilbert-type integral inequalities, Bentham Science Publishers Ltd., The United Emirates, 2009.
  • B. C. Yang, On Hilbert's integral inequality, J. Math. Anal. Appl. 220 (1998), no. 2, 778–785.
  • B. C. Yang, On an extension of Hilbert's integral inequality with some parameters, Aust. J. Math. Anal. Appl. 1 (2004), no. 1, Art. 11, 8 pp.
  • B. C. Yang, I. Brnetić, M. Krnić, and J. E. Pečarić, Generalization of Hilbert and Hardy–Hilbert integral inequalities, Math. Inequal. Appl. 8 (2005), no. 2, 259–272.