Skip to main content
SpringerLink
Log in

On a Hilbert-Type Integral Inequality in the Whole Plane Related to the Extended Riemann Zeta Function

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

In the present paper, a few equivalent conditions of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The best possible constant factor is related to the extended Riemann zeta function. In the form of applications, a few equivalent conditions of a Hilbert-type integral inequality with the homogeneous kernel in the whole plane are deduced. We also consider the operator expressions and a few particular cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934)

    MATH  Google Scholar 

  2. Yang, B.C.: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009)

    Google Scholar 

  3. Yang, B.C.: Hilbert-Type Integral Inequalities. Bentham Science, Sharjah (2009)

    Google Scholar 

  4. Yang, B.C.: On the norm of an integral operator and applications. J. Math. Anal. Appl. 321, 182–192 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Xu, J.S.: Hardy–Hilbert’s inequalities with two parameters. Adv. Math. 36(2), 63–76 (2007)

    MathSciNet  Google Scholar 

  6. Yang, B.C.: On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl. 325, 529–541 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Xin, D.M.: A Hilbert-type integral inequality with the homogeneous kernel of zero degree. Math. Theory Appl. 30(2), 70–74 (2010)

    MathSciNet  Google Scholar 

  8. Yang, B.C.: A Hilbert-type integral inequality with the homogenous kernel of degree 0. J. Shandong Univ. (Nat. Sci.) 45(2), 103–106 (2010)

    MathSciNet  Google Scholar 

  9. Debnath, L., Yang, B.C.: Recent developments of Hilbert-type discrete and integral inequalities with applications. Int. J. Math. Math. Sci. (2012). https://doi.org/10.1155/2012/871845

    MathSciNet  MATH  Google Scholar 

  10. Rassias, M.T., Yang, B.C.: On half-discrete Hilbert’s inequality. Appl. Math. Comput. 220, 75–93 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Yang, B.C., Krnic, M.: A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree 0. J. Math. Inequal. 6(3), 401–417 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Rassias, T.M., Yang, B.C.: A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function. Appl. Math. Comput. 225, 263–277 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Rassias, M.T., Yang, B.C.: On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic cotangent function. Appl. Math. Comput. 242, 800–813 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Rassias, M.T., Yang, B.C.: A multidimensional Hilbert-type integral inequality related to the Riemann zeta function. In: Daras, N.J. (ed.) Appl. Math. Inf. Sci. Eng., pp. 417–433. Springer, New York (2014)

    Google Scholar 

  15. Chen, Q., Yang, B.C.: A survey on the study of Hilbert-type inequalities. J. Inequal. Appl. 2015, 302 (2015)

    Article  MathSciNet  Google Scholar 

  16. Yang, B.C.: A new Hilbert-type integral inequality. Soochow J. Math. 33(4), 849–859 (2007)

    MathSciNet  MATH  Google Scholar 

  17. He, B., Yang, B.C.: On a Hilbert-type integral inequality with the homogeneous kernel of 0-degree and the hypergeometrc function. Math. Pract. Theory 40(18), 105–211 (2010)

    Google Scholar 

  18. Yang, B.C.: A new Hilbert-type integral inequality with some parameters. J. Jilin Univ. (Sci. Ed.) 46(6), 1085–1090 (2008)

    MathSciNet  Google Scholar 

  19. Yang, B.C.: A Hilbert-type integral inequality with a non-homogeneous kernel. J. Xiamen Univ. (Nat. Sci.) 48(2), 165–169 (2008)

    Google Scholar 

  20. Zeng, Z., Xie, Z.T.: On a new Hilbert-type integral inequality with the homogeneous kernel of degree 0 and the integral in whole plane. J. Inequal. Appl. (2010). https://doi.org/10.26634/jmat.1.1.1664

    Google Scholar 

  21. Yang, B.C.: A reverse Hilbert-type integral inequality with some parameters. J. Xinxiang Univ. (Nat. Sci. Ed.) 27(6), 1–4 (2010)

    Google Scholar 

  22. Wang, A.Z., Yang, B.C.: A new Hilbert-type integral inequality in whole plane with the non-homogeneous kernel. J. Inequal. Appl. 2011, 123 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Xin, D.M., Yang, B.C.: A Hilbert-type integral inequality in whole plane with the homogeneous kernel of degree-2. J. Inequal. Appl. (2011). https://doi.org/10.1155/2011/401428

    MathSciNet  MATH  Google Scholar 

  24. He, B., Yang, B.C.: On an inequality concerning a non-homogeneous kernel and the hypergeometric function. Tamsul Oxf. J. Inf. Math. Sci. 27(1), 75–88 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Yang, B.: A reverse Hilbert-type integral inequality with a non-homogeneous kernel. J. Jilin Univ. (Sci. Ed.) 49(3), 437–441 (2011)

    Google Scholar 

  26. Xie, Z.T., Zeng, Z., Sun, Y.F.: A new Hilbert-type inequality with the homogeneous kernel of degree-2. Adv. Appl. Math. Sci. 12(7), 391–401 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Huang, Q.L., Wu, S.H., Yang, B.C.: Parameterized Hilbert-type integral inequalities in the whole plane. Sci. World J. (2014). https://doi.org/10.1155/2014/169061

    Google Scholar 

  28. Zhen, Z., Raja Rama Gandhi, K., Xie, Z.T.: A new Hilbert-type inequality with the homogeneous kernel of degree-2 and with the integral. Bull. Math. Sci. Appl. 3(1), 11–20 (2014)

    Google Scholar 

  29. Rassias, M.T., Yang, B.C.: A Hilbert-type integral inequality in the whole plane related to the hyper geometric function and the beta function. J. Math. Anal. Appl. 428(2), 1286–1308 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Huang, X.Y., Cao, J.F., He, B., Yang, B.C.: Hilbert-type and Hardy-type integral inequalities with operator expressions and the best constants in the whole plane. J. Inequal. Appl. 2015, 129 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gu, Z.H., Yang, B.C.: A Hilbert-type integral inequality in the whole plane with a non-homogeneous kernel and a few parameters. J. Inequal. Appl. 2015, 314 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, Z.Q., Guo, D.R.: Introduction to Special Functions. Science Press, Beijing (1979)

    Google Scholar 

  33. Kuang, J.C.: Applied Inequalities. Shangdong Science and Technology Press, Jinan (2004)

    Google Scholar 

  34. Kuang, J.C.: Real and Functional Analysis (Continuation), vol. 2. Higher Education Press, Beijing (2015)

    Google Scholar 

Download references

Acknowledgements

M. Th. Rassias: I would like to express my gratitude to the J. S. Latsis Foundation for their financial support provided under the auspices of my current “Latsis Foundation Senior Fellowship” position. B. Yang: This work is supported by the National Natural Science Foundation (Nos. 61370186, 61640222), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25). I feel grateful for this help.

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Zurich, 8057, Zurich, Switzerland

    Michael Th. Rassias

  2. Moscow Institute of Physics and Technology, Institutskiy per, d. 9, Dolgoprudny, Russia, 141700

    Michael Th. Rassias

  3. Institute for Advanced Study, Program in Interdisciplinary Studies, 1 Einstein Dr, Princeton, NJ, 08540, USA

    Michael Th. Rassias

  4. Department of Mathematics, Guangdong University of Education, Guangzhou, 510303, Guangdong, People’s Republic of China

    Bicheng Yang

Authors
  1. Michael Th. Rassias
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bicheng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Th. Rassias.

Additional information

Communicated by Daniel Aron Alpay.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rassias, M.T., Yang, B. On a Hilbert-Type Integral Inequality in the Whole Plane Related to the Extended Riemann Zeta Function. Complex Anal. Oper. Theory 13, 1765–1782 (2019). https://doi.org/10.1007/s11785-018-0830-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-018-0830-5

Keywords

Mathematics Subject Classification

Search

Navigation