A Multidimensional Hilbert-Type Integral Inequality Related to the Riemann Zeta Function

  • Michael Th. RassiasEmail author
  • Bicheng Yang
Part of the Springer Optimization and Its Applications book series (SOIA, volume 91)


In this chapter, using methods of weight functions and techniques of real analysis, we provide a multidimensional Hilbert-type integral inequality with a homogeneous kernel of degree 0 as well as a best possible constant factor related to the Riemann zeta function. Some equivalent representations and certain reverses are obtained. Furthermore, we also consider operator expressions with the norm and some particular results.


Hilbert-type integral inequality Hilbert-type integral operator Riemann zeta function Gamma function Weight function 



The authors wish to express their thanks to Professors Tserendorj Batbold, Mario Krnic, and Jichang Kuang for their careful reading of the manuscript and for their valuable suggestions.

M. Th. Rassias: This work is supported by the Greek State Scholarship Foundation (IKY).

B. Yang: This work is supported by 2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University (No. 2012KJCX0079).


  1. 1.
    Hardy G. H., Littlewood J. E., Pólya G., Inequalities. Cambridge University Press, Cambridge, 1934Google Scholar
  2. 2.
    Mitrinović D. S., Pečarić J. E., Fink A. M., Inequalities Involving Functions and their Integrals and Derivatives. Kluwer Acaremic Publishers, Boston, 1991CrossRefzbMATHGoogle Scholar
  3. 3.
    Batbold Ts., Adiyasuren V., Castillo R. E., Extension of reverse Hilbert-type inequality with a generalized homogeneous kernel. Rev. Colombiana Mat. 45(2), 187-195 (2011)Google Scholar
  4. 4.
    Yang B. C., Hilbert-Type Integral Inequalities. Bentham Science Publishers Ltd., Sharjah, 2009Google Scholar
  5. 5.
    Yang B. C., Discrete Hilbert-Type Inequalities. Bentham Science Publishers Ltd., Sharjah, 2011Google Scholar
  6. 6.
    Yang B. C., The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing, 2009, ChinaGoogle Scholar
  7. 7.
    Yang B. C., Hilbert-type Integral operators: norms and inequalities, In: Nonlinear Analysis, Stability, Approximation, and Inequalities (P. M. Pardalos, et al.). Springer, New York, 2012, pp. 771–859Google Scholar
  8. 8.
    Yang B. C., On Hilbert’s integral inequality. Journal of Mathematical Analysis and Applications, 220, 778–785(1998)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Yang B. C., Brnetić I, Krnić M., Pe\(\check{c}\) arić J. E., Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Ineq. and Appl., 8(2), 259–272 (2005)Google Scholar
  10. 10.
    Krnić M., Pe\(\check{c}\)arić J. E., Hilbert’s inequalities and their reverses, Publ. Math. Debrecen, 67(3–4), 315–331 (2005)Google Scholar
  11. 11.
    Rassias M. Th., Yang B. C., On half - discrete Hilbert’s inequality, Applied Mathematics and Computation, to appearGoogle Scholar
  12. 12.
    Azar L., On some extensions of Hardy-Hilbert’s inequality and Applications. Journal of Inequalities and Applications, 2009, No. 546829Google Scholar
  13. 13.
    Arpad B., Choonghong O., Best constant for certain multilinear integral operator. Journal of Inequalities and Applications, 2006, No. 28582Google Scholar
  14. 14.
    Kuang J. C., Debnath L., On Hilbert’s type inequalities on the weighted Orlicz spaces. Pacific J. Appl. Math., 1(1), 95–103 (2007)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Zhong W. Y., The Hilbert-type integral inequality with a homogeneous kernel of −λ-degree. Journal of Inequalities and Applications, 2008, No. 917392Google Scholar
  16. 16.
    Hong Y., On Hardy-Hilbert integral inequalities with some parameters. J. Ineq. in Pure & Applied Math.,6(4), Art. 92, 1–10 (2005)Google Scholar
  17. 17.
    Zhong W. Y., Yang B. C., On multiple Hardy-Hilbert’s integral inequality with kernel. Journal of Inequalities and Applications, Vol. 2007, Art.ID 27962, 17 pages, doi: 10.1155/ 2007/27Google Scholar
  18. 18.
    Yang B. C., Krnić M., On the norm of a multidimensional Hilbert-type operator, Sarajevo Journal of Mathematics, 7(20), 223–243(2011)Google Scholar
  19. 19.
    Li Y. J., He B., On inequalities of Hilbert’s type. Bulletin of the Australian Mathematical Society, 76(1), 1–13 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Edwards H. M., Riemann’s Zeta Function. Dover Publications, New York, 1974zbMATHGoogle Scholar
  21. 21.
    Milovanović G. V., Rassias M. Th. (eds.), Analytic Number Theory, Approximation Theory and Special Functions. Springer, New York, to appearGoogle Scholar
  22. 22.
    Apostol T. M., Introduction to Analytic Number Theory. Springer – Verlag, New York, 1984Google Scholar
  23. 23.
    Erdős P., Suranyi J., Topics in the Theory of Numbers. Springer – Verlag, New York, 2003CrossRefGoogle Scholar
  24. 24.
    Hardy G. H., Wright E. W., An Introduction to the Theory of Numbers. 5th edition, Clarendon Press, Oxford, 1979zbMATHGoogle Scholar
  25. 25.
    Iwaniec H., Kowalski E., Analytic Number Theory. American Mathematical Society, Colloquium Publications, Volume 53, Rhode Island, 2004Google Scholar
  26. 26.
    Landau E., Elementary Number Theory. 2nd edition, Chelsea, New York, 1966Google Scholar
  27. 27.
    Miller S. J., Takloo – Bighash R., An Invitation to Modern Number Theory. Princeton University Press, Princeton and Oxford, 2006Google Scholar
  28. 28.
    Zhong Y. Q., On Complex Functions. Higher Education Press, Beijing, China, 2004Google Scholar
  29. 29.
    Kuang J. C., Introduction to Real Analysis. Hunan Education Press, Chansha, China, 1996Google Scholar
  30. 30.
    Kuang J. C., Applied Inequalities. Shangdong Science Technic Press, Jinan, China, 2004Google Scholar
  31. 31.
    Rassias M. Th., Problem–Solving and Selected Topics in Number Theory: In the Spirit of the Mathematical Olympiads (Foreword by Preda Mihăilescu). Springer, New York, 2011CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsETH-ZentrumZurichSwitzerland
  2. 2.Department of MathematicsGuangdong University of EducationGuangzhouP. R. China

Personalised recommendations