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On a Hilbert-Type Integral Inequality in the Whole Plane

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Applications of Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 134))

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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.

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References

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  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. M.Th. Rassias, B.C. Yang, On a half-discrete Hilbert’s inequality. Appl. Math. Comput. 220, 75–93 (2013)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. 417–433

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. 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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  33. J.C. Kuang, Real Analysis and Functional Analysis (Continuation) (Second Volume) (Higher Education Press, Beijing, 2015)

    Google Scholar 

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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.

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Correspondence to Bicheng Yang .

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Rassias, M.T., Yang, B. (2018). On a Hilbert-Type Integral Inequality in the Whole Plane. In: Rassias, T. (eds) Applications of Nonlinear Analysis . Springer Optimization and Its Applications, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-319-89815-5_23

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