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On a Hilbert-Type Integral Inequality in the Whole Plane

  • Michael Th. Rassias
  • Bicheng YangEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 134)

Abstract

By using methods of real analysis and weight functions, we prove a new Hilbert-type integral inequality in the whole plane with non-homogeneous kernel and a best possible constant factor. As applications, we also consider the equivalent forms, some particular cases and the operator expressions.

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation (No. 61772140), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25). We are grateful for their help.

References

  1. 1.
    G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, Cambridge, 1934)zbMATHGoogle Scholar
  2. 2.
    B.C. Yang, The Norm of Operator and Hilbert-Type Inequalities (Science Press, Beijing, 2009)Google Scholar
  3. 3.
    B.C. Yang, Hilbert-Type Integral Inequalities (Bentham Science Publishers Ltd., Sharjah, 2009)Google Scholar
  4. 4.
    B.C. Yang, On the norm of an integral operator and applications. J. Math. Anal. Appl. 321, 182–192 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    J.S. Xu, Hardy-Hilbert’s inequalities with two parameters. Adv. Math. 36(2), 63–76 (2007)MathSciNetGoogle Scholar
  6. 6.
    B.C. Yang, On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl. 325, 529–541 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D.M. Xin, A Hilbert-type integral inequality with the homogeneous kernel of zero degree. Math. Theory Appl. 30(2), 70–74 (2010)MathSciNetGoogle Scholar
  8. 8.
    B.C. Yang, A Hilbert-type integral inequality with the homogenous kernel of degree 0. J. Shandong Univ. 45(2), 103–106 (2010)MathSciNetGoogle Scholar
  9. 9.
    L. Debnath, B.C. Yang, Recent developments of Hilbert-type discrete and integral inequalities with applications. Int. J. Math. Math. Sci. 2012, 871845, 29 (2012)Google Scholar
  10. 10.
    M.Th. Rassias, B.C. Yang, On a half-discrete Hilbert’s inequality. Appl. Math. Comput. 220, 75–93 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    B.C. Yang, M. Krnić, A half-discrete Hilbert-type inequality with a general homogeneous kernel of degree 0. J. Math. Inequal. 6(3), 401–417 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Th.M. Rassias, B.C. Yang, A multidimensional half - discrete Hilbert - type inequality and the Riemann zeta function. Appl. Math. Comput. 225, 263–277 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    M.Th. Rassias, B.C. Yang, On a multidimensional half - discrete Hilbert - type inequality related to the hyperbolic cotangent function. Appl. Math. Comput. 242, 800–813 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    M.Th. Rassias, B.C. Yang, A multidimensional Hilbert - type integral inequality related to the Riemann zeta function, in Applications of Mathematics and Informatics in Science and Engineering, ed. by N.J. Daras (Springer, New York, 2014), pp. 417–433CrossRefGoogle Scholar
  15. 15.
    Q. Chen, B.C. Yang, A survey on the study of Hilbert-type inequalities. J. Inequal. Appl. 2015, 302 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    B.C. Yang, A new Hilbert-type integral inequality. Soochow J. Math. 33(4), 849–859 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Z.Q. Wang, D.R. Guo, Introduction to Special Functions (Science Press, Beijing, 1979)Google Scholar
  18. 18.
    B. He, B.C. Yang, 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
  19. 19.
    B.C. Yang, A new Hilbert-type integral inequality with some parameters. J. Jilin Univ. 46(6), 1085–1090 (2008)MathSciNetGoogle Scholar
  20. 20.
    B.C. Yang, A Hilbert-type integral inequality with a non-homogeneous kernel. J. Xiamen Univ. 48(2), 165–169 (2008)Google Scholar
  21. 21.
    Z. Zeng, Z.T. Xie, On a new Hilbert-type integral inequality with the homogeneous kernel of degree 0 and the integral in whole plane. J. Inequal. Appl. 2010, 256796, 9 (2010)Google Scholar
  22. 22.
    B.C. Yang, A reverse Hilbert-type integral inequality with some parameters. J. Xinxiang Univ. 27(6), 1–4 (2010)MathSciNetGoogle Scholar
  23. 23.
    A.Z. Wang, B.C. Yang, A new Hilbert-type integral inequality in whole plane with the non-homogeneous kernel. J. Inequal. Appl. 2011, 123 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    D.M. Xin, B.C. Yang, A Hilbert-type integral inequality in whole plane with the homogeneous kernel of degree -2. J. Inequal. Appl. 2011, 401428, 11 (2011)Google Scholar
  25. 25.
    B. He, B.C. Yang, On an inequality concerning a non-homogeneous kernel and the hypergeometric function. Tamsul Oxford J. Inf. Math. Sci. 27(1), 75–88 (2011)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Z.T. Xie, Z. Zeng, Y.F. Sun, A new Hilbert-type inequality with the homogeneous kernel of degree -2. Adv. Appl. Math. Sci. 12(7), 391–401 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Q.L. Huang, S.H. Wu, B.C. Yang, Parameterized Hilbert-type integral inequalities in the whole plane. Sci. World J. 2014, 169061, 8 (2014)Google Scholar
  28. 28.
    Z. Zhen, K. Raja Rama Gandhi, Z.T. Xie, 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. 29.
    M.Th. Rassias, B.C. Yang, 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)MathSciNetCrossRefGoogle Scholar
  30. 30.
    X.Y. Huang, J.F.Cao, B. He, B.C. Yang, Hilbert-type and Hardy-type integral inequalities with operator expressions and the best constants in the whole plane. J. Inequal. Appl. 2015, 129 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Z.H. Gu, B.C. Yang, A Hilbert-type integral inequality in the whole plane with a non-homogeneous kernel and a few parameters. J. Inequal. Appl. 2015, 314 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    J.C. Kuang, Applied Inequalities (Shangdong Science and Technology Press, Jinan, 2004)Google Scholar
  33. 33.
    J.C. Kuang, Real Analysis and Functional Analysis (Continuation) (Second Volume) (Higher Education Press, Beijing, 2015)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of ZurichZurichSwitzerland
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  3. 3.Institute for Advanced Study, Program in Interdisciplinary StudiesPrincetonUSA
  4. 4.Department of MathematicsGuangdong University of EducationGuangzhou, GuangdongPeople’s Republic of China

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