Skip to main content
SpringerLink
Log in

Hilbert-Type Integral Operators: Norms and Inequalities

  • Chapter
Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 68))

Abstract

The well known Hilbert inequality and Hardy–Hilbert inequality may be rewritten in the forms of inequalities relating Hilbert operator and Hardy–Hilbert operator with their norms. These two operators are some particular kinds of Hilbert-type operators, which have played an important role in mathematical analysis and applications. In this chapter, by applying the methods of Real Analysis and Operator Theory, we define a general Hilbert-type integral operator and study six particular kinds of this operator with different measurable kernels in several normed spaces. The norms, equivalent inequalities, some particular examples, and compositions of two operators are considered. In Sect. 42.1, we define the weight functions with some parameters and give two equivalent inequalities with the general measurable kernels. Meanwhile, the norm of a Hilbert-type integral operator is estimated. In Sect. 42.2 and Sect. 42.3, four kinds of Hilbert-type integral operators with the particular kernels in the first quarter and in the whole plane are obtained. In Sect. 42.4, we define two kinds of operators with the kernels of multi-variables and obtain their norms. In Sect. 42.5, two kinds of compositions of Hilbert-type integral operators are considered. The lemmas and theorems provide an extensive account for this kind of operators.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Kuang, J.C.: Applied Inequalities, 3nd edn. Shangdong Science Technic Press, Jinan (2004)

    Google Scholar 

  2. Kuang, J.C.: Introduction to Real Analysis. Hunan Education Press, Chansha (1996)

    Google Scholar 

  3. Wilhelm, M.: On the spectrum of Hilbert’s matrix. Am. J. Math. 72, 699–704 (1950)

    Article  MATH  Google Scholar 

  4. Carleman, T.: Sur les equations integrals singulieres a noyau reel et symetrique. Uppsala (1923)

    Google Scholar 

  5. Zang, K.W.: A bilinear inequality. J. Math. Anal. Appl. 271, 288–296 (2002)

    Article  MathSciNet  Google Scholar 

  6. 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 

  7. Yang, B.C.: On the norm of a self-adjoint operator and a new bilinear integral inequality. Acta Math. Sin. Engl. Ser. 23(7), 1311–1316 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Yang, B.C.: On the norm of a certain self-adjoint integral operator and applications to bilinear integral inequalities. Taiwan. J. Math. 12(2), 315–324 (2008)

    MATH  Google Scholar 

  9. 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 

  10. Yang, B.C.: On the norm of a self-adjoint operator and application to Hilbert’s type inequalities. Bull. Belg. Math. Soc. 13, 577–584 (2006)

    MATH  Google Scholar 

  11. Yang, B.C.: On a Hilbert-type operator with a symmetric homogeneous kernel of −1-degree and application. Arch. Inequal. Appl. (2007). doi:10.1155/2007/47812

    Google Scholar 

  12. Yang, B.C.: On the norm of a linear operator and its applications. Indian J. Pure Appl. Math. 39(3), 237–250 (2008)

    MathSciNet  Google Scholar 

  13. Yang, B.C.: On a Hilbert-type operator with a class of homogeneous kernel. Arch. Inequal. Appl. (2009). doi:10.1155/2009/572176

    Google Scholar 

  14. Yang, B.C.: A new Hilbert-type operator and applications. Publ. Math. (Debr.) 76(1–2), 147–156 (2010)

    MATH  Google Scholar 

  15. Yang, B.C., Rassias, T.M.: On a Hilbert-type integral inequality in the subinterval and its operator expression. Banach J. Math. Anal. 4(2), 100–110 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Huang, Q.L., Yang, B.C.: On a multiple Hilbert-type integral operator and applications. Arch. Inequal. Appl. (2009). doi:10.1155/2009/192197

    Google Scholar 

  17. Arpad, B., Choonghong, O.: Best constants for certain multilinear integral operator. Arch. Inequal. Appl. (2006). doi:10.1155/2006/28582

    Google Scholar 

  18. Zhong, W.Y.: A Hilbert-type linear operator with the norm and its applications. Arch. Inequal. Appl. (2009). doi:10.1155/2009/494257

    Google Scholar 

  19. Zhong, W.Y.: A new Hilbert-type linear operator with the a composite kernel and its applications. Arch. Inequal. Appl. (2010). doi:10.1155/2010/393025

    Google Scholar 

  20. Li, Y.J., He, B.: Hilbert’s type linear operator and some extensions of Hilbert’s inequality. Arch. Inequal. Appl. (2007). doi:10.1155/2007/82138

    Google Scholar 

  21. Liu, X.D., Yang, B.C.: On a Hilbert–Hardy-type integral operator and applications. Arch. Inequal. Appl. (2010). doi:10.1155/2010/812636

    Google Scholar 

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

    Google Scholar 

  23. Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston (1991)

    Book  MATH  Google Scholar 

  24. Yang, B.C., Gao, M.Z.: On a best value of Hardy–Hilbert’s inequality. Adv. Math. 26(2), 159–164 (1997)

    MathSciNet  MATH  Google Scholar 

  25. Gao, M.Z., Yang, B.C.: On the extended Hilbert’s inequality. Proc. Am. Math. Soc. 126(3), 751–759 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pachpatte, B.G.: On some new inequalities similar to Hilbert’s inequality. J. Math. Anal. Appl. 226, 166–179 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yang, B.C.: On Hilbert’s integral inequality. J. Math. Anal. Appl. 220, 778–785 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yang, B.C., Debnath, L.: On a new generalization of Hardy-Hilbert’s inequality. J. Math. Anal. Appl. 233, 484–497 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, B.C., Rassias, T.M.: On the way of weight coefficient and research for Hilbert-type inequalities. Math. Inequal. Appl. 6(4), 625–658 (2003)

    MathSciNet  MATH  Google Scholar 

  30. Yang, B.C.: On new extension of Hilbert’s inequality. Acta Math. Hung. 104(4), 291–299 (2004)

    Article  MATH  Google Scholar 

  31. Yang, B.C.: On an extension of Hilbert’s integral inequality with some parameters. Aust. J. Math. Anal. Appl. 1(1), 11 (2004), pp. 8

    MathSciNet  Google Scholar 

  32. Yang, B.C., Brnete, I., Krnic, M., et al.: Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Inequal. Appl. 8(2), 259–272 (2005)

    MathSciNet  MATH  Google Scholar 

  33. Hu, K.: Some Problems in Analysis Inequalities. Wuhan University Press, Wuhan (2007)

    Google Scholar 

  34. Yang, B.C.: A survey of the study of Hilbert-type inequalities with parameters. Adv. Math. 38(3), 257–268 (2009)

    MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Google Scholar 

  37. Yang, B.C.: Discrete Hilbert-Type Inequalities. Bentham Science Publishers Ltd. (2011)

    Google Scholar 

  38. Wang, D.X., Guo, D.R.: Special Functions. Science Press, Beijing (1979)

    Google Scholar 

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

    MATH  Google Scholar 

  40. Yang, B.C.: A Hilbert-type inequality with a non-homogeneous kernel. J. Xiamen University (Nat. Sci.) 48(3), 165–169 (2009)

    Google Scholar 

  41. Ping, Y., Wang, H., Song, L.: Complex Functions. Science Press, Beijing (2004)

    Google Scholar 

  42. 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. Arch. Inequal. Appl. (2010). doi:10.1155/2010/256796

    Google Scholar 

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

    Google Scholar 

  44. Zhong, W.Y., Yang, B.C.: On multiple’s Hardy-Hilbert integral inequality with kernel. Arch. Inequal. Appl. (2007). doi:10.1155/2007/27962

    Google Scholar 

  45. Hong, Y.: On multiple Hardy-Hilbert integral inequalities with some parameters. Arch. Inequal. Appl. (2002). doi:10.1155/2006/94960

    Google Scholar 

  46. Yang, B.C.: On an application of Hilbert’s inequality with multi-parameters. J. Beijing Union University (Nat. Sci.) 24(4), 78–84 (2010)

    Google Scholar 

  47. Yang, B.C.: An application of the reverse Hilbert’s inequality with multi-parameters. J. Xinxiang University (Nat. Sci.) 27(4), 1–5 (2010)

    MATH  Google Scholar 

  48. Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis. Wiley, New York (1980)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong, 510303, P.R. China

    Bicheng Yang

Authors
  1. Bicheng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bicheng Yang .

Editor information

Editors and Affiliations

  1. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

  2. Center for Applied Optimization, Dept. Industrial & Systems, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Pando G. Georgiev

  3. , Department of Mathematics & Statistics, University of Victoria, Finnerty Road 3800, Victoria, V8W 3R4, British Columbia, Canada

    Hari M. Srivastava

Additional information

Dedicated to Professor Themistocles M. Rassias on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Yang, B. (2012). Hilbert-Type Integral Operators: Norms and Inequalities. In: Pardalos, P., Georgiev, P., Srivastava, H. (eds) Nonlinear Analysis. Springer Optimization and Its Applications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3498-6_42

Download citation

Publish with us

Policies and ethics

Search

Navigation