Advertisement

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

  • Michael Th. Rassias
  • Bicheng Yang
Article
  • 22 Downloads

Abstract

By using techniques of real analysis and weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The constant factor related the extended Hurwitz zeta function is proved to be the best possible. As applications, a few equivalent statements of a Hilbert-type integral inequality with the homogeneous kernel in the whole plane are deduced. We also consider the operator expressions and some corollaries.

Keywords

Hilbert-type integral inequality Weight function Equivalent form Operator Hurwitz zeta function 

Mathematics Subject Classification

26D15 47A07 65B10 

Notes

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 and 61640222), and Appropriative Researching Fund for Professors and Doctors, Guangdong University of Education (No. 2015ARF25). I am grateful for this help.

References

  1. 1.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934) zbMATHGoogle Scholar
  2. 2.
    Yang, B.C.: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing (2009) Google Scholar
  3. 3.
    Yang, B.C.: Hilbert-Type Integral Inequalities. Bentham Science, Emirate of Sharjah (2009) Google Scholar
  4. 4.
    Yang, B.C.: On the norm of an integral operator and applications. J. Math. Anal. Appl. 321, 182–192 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Xu, J.S.: Hardy-Hilbert’s inequalities with two parameters. Adv. Math. 36(2), 63–76 (2007) MathSciNetGoogle Scholar
  6. 6.
    Yang, B.C.: On the norm of a Hilbert’s type linear operator and applications. J. Math. Anal. Appl. 325, 529–541 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Xin, D.M.: A Hilbert-type integral inequality with the homogeneous kernel of zero degree. Math. Theory Appl. 30(2), 70–74 (2010) MathSciNetGoogle Scholar
  8. 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) MathSciNetGoogle Scholar
  9. 9.
    Debnath, L., Yang, B.C.: Recent developments of Hilbert-type discrete and integral inequalities with applications. Int. J. Math. Math. Sci. 2012, 871845 (2012), 29 pages MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hong, Y.: On the structure character of Hilbert’s type integral inequality with homogeneous kernel and applications. J. Jilin Univ. Sci. Ed. 55(2), 189–194 (2017) Google Scholar
  11. 11.
    Rassias, M.Th., Yang, B.C.: On half-discrete Hilbert’s inequality. Appl. Math. Comput. 220, 75–93 (2013) MathSciNetzbMATHGoogle Scholar
  12. 12.
    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) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Rassias, Th.M., Yang, B.C.: A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function. Appl. Math. Comput. 225, 263–277 (2013) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Rassias, M.Th., Yang, B.C.: On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic cotangent function. Appl. Math. Comput. 242, 800–813 (2013) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Rassias, M.Th., Yang, B.C.: A multidimensional Hilbert-type integral inequality related to the Riemann zeta function. In: Daras, N.J. (ed.) Applications of Mathematics and Informatics in Science and Engineering, pp. 417–433. Springer, New York (2014) CrossRefGoogle Scholar
  16. 16.
    Chen, Q., Yang, B.C.: A survey on the study of Hilbert-type inequalities. J. Inequal. Appl. 2015, 302 (2015) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yang, B.C.: A new Hilbert-type integral inequality. Soochow J. Math. 33(4), 849–859 (2007) MathSciNetzbMATHGoogle Scholar
  18. 18.
    He, B., Yang, B.C.: On a Hilbert-type integral inequality with the homogeneous kernel of 0-degree and the hypergeometric function. Math. Pract. Theory 40(18), 105–211 (2010) MathSciNetGoogle Scholar
  19. 19.
    Wang, Z.Q., Guo, D.R.: Introduction to Special Functions. Science Press, Beijing (1979) Google Scholar
  20. 20.
    Yang, B.C.: A new Hilbert-type integral inequality with some parameters. J. Jilin Univ. Sci. Ed. 46(6), 1085–1090 (2008) MathSciNetGoogle Scholar
  21. 21.
    Yang, B.C.: A Hilbert-type integral inequality with a non-homogeneous kernel. J. Xiamen Univ. Natur. Sci. 48(2), 165–169 (2008) Google Scholar
  22. 22.
    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, 256796 (2010), 9 pages zbMATHCrossRefGoogle Scholar
  23. 23.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    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, 401428 (2011), 11 pages MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    He, B., Yang, B.C.: On an inequality concerning a non-homogeneous kernel and the hypergeometric function. Tamsui Oxf. J. Inf. Math. Sci. 27(1), 75–88 (2011) MathSciNetzbMATHGoogle Scholar
  26. 26.
    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
  27. 27.
    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) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Huang, Q.L., Wu, S.H., Yang, B.C.: Parameterized Hilbert-type integral inequalities in the whole plane. Sci. World J. 2014, 169061 (2014), 8 pages Google Scholar
  29. 29.
    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
  30. 30.
    Rassias, M.Th., 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) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Kuang, J.C.: Real and Functional Analysis (Continuation), vol. 2. Higher Education Press, Beijing (2015) Google Scholar
  34. 34.
    Kuang, J.C.: Applied Inequalities. Shangdong Science and Technology Press, Jinan (2004) Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of ZurichZurichSwitzerland
  2. 2.Institute for Advanced Study Program in Interdisciplinary StudiesPrincetonUSA
  3. 3.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  4. 4.Department of MathematicsGuangdong University of EducationGuangzhouP.R. China

Personalised recommendations