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Year 2025, Volume: 8 Issue: 2, 93 - 116, 15.06.2025
https://doi.org/10.33205/cma.1698981

Abstract

References

  • V. Adiyasuren, T. Batbold and M. Krni´c: Multiple Hilbert-type inequalities involving some differential operators, Banach J. Math. Anal., 10 (2016), 320–337.
  • L. E. Azar: The connection between Hilbert and Hardy inequalities, J. Inequal. Appl., 2013 (2013), 1–10.
  • A. Bényi, C. T. Oh: Best constant for certain multilinear integral operator, J. Inequal. Appl., 2006 (2006), 1–12.
  • C. Chesneau: Some four-parameter trigonometric generalizations of the Hilbert integral inequality, Asia Math., 8 (2) (2024), 45–59.
  • C. Chesneau: Two new general integral results related to the Hilbert integral inequality, Fundam. J. Math. Appl., 8 (5) (2025), 1-–11.
  • I. S. Gradshteyn and I. M. Ryzhik: Table of Integrals, Series, and Products, 7th Edition, Academic Press (2007).
  • G. H. Hardy, J. E. Littlewood and G. Polya: Inequalities, Cambridge University Press, Cambridge (1952).
  • S. W. Jian, F. Z. Yang: All-sided generalization about Hardy-Hilbert integral inequalities, Acta Math. Sin. Chin., 44 (4) (2001), 619–626.
  • Y. Li, Y. Qian and B. He: On further analogs of Hilbert’s inequality, Int. J. Math. Sci., 2007 (2007), 1–6.
  • Y. Li, J.Wu and B. He: A new Hilbert-type integral inequality and the equivalent form, Int. J. Math. Math. Sci., 18 (2006), 1–6.
  • E. M. Stein, R. Shakarchi: Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, Princeton (2011).
  • W. T. Sulaiman: On Hardy-Hilbert’s integral inequality, J. Inequal. Pure and Appl. Math., 5 (2) (2004), 1–9.
  • W. T. Sulaiman: On three inequalities similar to Hardy-Hilbert’s integral inequality, Acta. Math. Univ. Comenianae, IXXVI (2007), 273–278.
  • W. T. Sulaiman: New Hardy-Hilbert’s-type integral inequalities, Int. Math. Forum, 3 (2008), 2139–2147.
  • W. T. Sulaiman: New types of Hardy-Hilbert’s integral inequality, Gen. Math. Notes, 2 (2) (2011), 111–118.
  • B. Sun: A multiple Hilbert-type integral inequality with the best constant factor, J. Inequal. Appl., 2007 (2007), 1–14.
  • Z. T. Xie, Z. Zeng and Y. F. Sun: A new Hilbert-type inequality with the homogeneous kernel of degree -2, Adv. Appl. Math. Sci., 12 (7) (2013), 391–401.
  • J. S. Xu: Hardy-Hilbert’s inequalities with two parameters, Adv. Math., 36 (2) (2007), 63–76.
  • B. C. Yang: On Hilbert’s integral inequality, J. Math. Anal. Appl., 220 (2) (1998), 778–785.
  • B. C. Yang: On the norm of an integral operator and applications, J. Math. Anal. Appl., 321 (1) (2006), 182–192.
  • B. C. Yang: On the norm of a Hilbert’s type linear operator and applications, J. Math. Anal. Appl., 325 (2007), 529–541.
  • B. C. Yang: The Norm of Operator and Hilbert-Type Inequalities, Science Press, Beijing (2009).
  • B. C. Yang: Hilbert-Type Integral Inequalities, Bentham Science Publishers, The United Arab Emirates (2009).
  • B. C. Yang: Hilbert-type integral inequality with non-homogeneous kernel, J. Shanghai Univ., 17 (2011), 603–605.
  • B. C. Yang, I. Brnetic, M. Krnic and J. Pecaric: Generalization of Hilbert and Hardy-Hilbert integral inequalities, Math. Inequal. Appl., 8 (2) (2005), 259–272.

Study of some new one-parameter modifications of the Hardy-Hilbert integral inequality

Year 2025, Volume: 8 Issue: 2, 93 - 116, 15.06.2025
https://doi.org/10.33205/cma.1698981

Abstract

One of the pillars of mathematical analysis is the Hardy-Hilbert integral inequality. In this article, we advance the theory by introducing several new modifications of this inequality. They have the property of incorporating an adjustable parameter and different power functions, allowing for greater flexibility and broader applicability. Notably, one modification has a logarithmic structure, offering a distinctive extension to the classical framework. For the main results, the optimality of the corresponding constant factors is shown. Additional integral inequalities of various forms and scopes are also established. Thus, this work contributes to the ongoing development of Hardy-Hilbert-type inequalities by presenting new generalizations and providing rigorous mathematical justification for each result.

References

  • V. Adiyasuren, T. Batbold and M. Krni´c: Multiple Hilbert-type inequalities involving some differential operators, Banach J. Math. Anal., 10 (2016), 320–337.
  • L. E. Azar: The connection between Hilbert and Hardy inequalities, J. Inequal. Appl., 2013 (2013), 1–10.
  • A. Bényi, C. T. Oh: Best constant for certain multilinear integral operator, J. Inequal. Appl., 2006 (2006), 1–12.
  • C. Chesneau: Some four-parameter trigonometric generalizations of the Hilbert integral inequality, Asia Math., 8 (2) (2024), 45–59.
  • C. Chesneau: Two new general integral results related to the Hilbert integral inequality, Fundam. J. Math. Appl., 8 (5) (2025), 1-–11.
  • I. S. Gradshteyn and I. M. Ryzhik: Table of Integrals, Series, and Products, 7th Edition, Academic Press (2007).
  • G. H. Hardy, J. E. Littlewood and G. Polya: Inequalities, Cambridge University Press, Cambridge (1952).
  • S. W. Jian, F. Z. Yang: All-sided generalization about Hardy-Hilbert integral inequalities, Acta Math. Sin. Chin., 44 (4) (2001), 619–626.
  • Y. Li, Y. Qian and B. He: On further analogs of Hilbert’s inequality, Int. J. Math. Sci., 2007 (2007), 1–6.
  • Y. Li, J.Wu and B. He: A new Hilbert-type integral inequality and the equivalent form, Int. J. Math. Math. Sci., 18 (2006), 1–6.
  • E. M. Stein, R. Shakarchi: Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press, Princeton (2011).
  • W. T. Sulaiman: On Hardy-Hilbert’s integral inequality, J. Inequal. Pure and Appl. Math., 5 (2) (2004), 1–9.
  • W. T. Sulaiman: On three inequalities similar to Hardy-Hilbert’s integral inequality, Acta. Math. Univ. Comenianae, IXXVI (2007), 273–278.
  • W. T. Sulaiman: New Hardy-Hilbert’s-type integral inequalities, Int. Math. Forum, 3 (2008), 2139–2147.
  • W. T. Sulaiman: New types of Hardy-Hilbert’s integral inequality, Gen. Math. Notes, 2 (2) (2011), 111–118.
  • B. Sun: A multiple Hilbert-type integral inequality with the best constant factor, J. Inequal. Appl., 2007 (2007), 1–14.
  • Z. T. Xie, Z. Zeng and Y. F. Sun: A new Hilbert-type inequality with the homogeneous kernel of degree -2, Adv. Appl. Math. Sci., 12 (7) (2013), 391–401.
  • J. S. Xu: Hardy-Hilbert’s inequalities with two parameters, Adv. Math., 36 (2) (2007), 63–76.
  • B. C. Yang: On Hilbert’s integral inequality, J. Math. Anal. Appl., 220 (2) (1998), 778–785.
  • B. C. Yang: On the norm of an integral operator and applications, J. Math. Anal. Appl., 321 (1) (2006), 182–192.
  • B. C. Yang: On the norm of a Hilbert’s type linear operator and applications, J. Math. Anal. Appl., 325 (2007), 529–541.
  • B. C. Yang: The Norm of Operator and Hilbert-Type Inequalities, Science Press, Beijing (2009).
  • B. C. Yang: Hilbert-Type Integral Inequalities, Bentham Science Publishers, The United Arab Emirates (2009).
  • B. C. Yang: Hilbert-type integral inequality with non-homogeneous kernel, J. Shanghai Univ., 17 (2011), 603–605.
  • B. C. Yang, I. Brnetic, M. Krnic and J. Pecaric: Generalization of Hilbert and Hardy-Hilbert integral inequalities, Math. Inequal. Appl., 8 (2) (2005), 259–272.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Methods and Special Functions
Journal Section Articles
Authors

Christophe Chesneau 0000-0002-1522-9292

Early Pub Date June 6, 2025
Publication Date June 15, 2025
Submission Date May 14, 2025
Acceptance Date June 4, 2025
Published in Issue Year 2025 Volume: 8 Issue: 2

Cite

APA Chesneau, C. (2025). Study of some new one-parameter modifications of the Hardy-Hilbert integral inequality. Constructive Mathematical Analysis, 8(2), 93-116. https://doi.org/10.33205/cma.1698981
AMA Chesneau C. Study of some new one-parameter modifications of the Hardy-Hilbert integral inequality. CMA. June 2025;8(2):93-116. doi:10.33205/cma.1698981
Chicago Chesneau, Christophe. “Study of Some New One-Parameter Modifications of the Hardy-Hilbert Integral Inequality”. Constructive Mathematical Analysis 8, no. 2 (June 2025): 93-116. https://doi.org/10.33205/cma.1698981.
EndNote Chesneau C (June 1, 2025) Study of some new one-parameter modifications of the Hardy-Hilbert integral inequality. Constructive Mathematical Analysis 8 2 93–116.
IEEE C. Chesneau, “Study of some new one-parameter modifications of the Hardy-Hilbert integral inequality”, CMA, vol. 8, no. 2, pp. 93–116, 2025, doi: 10.33205/cma.1698981.
ISNAD Chesneau, Christophe. “Study of Some New One-Parameter Modifications of the Hardy-Hilbert Integral Inequality”. Constructive Mathematical Analysis 8/2 (June 2025), 93-116. https://doi.org/10.33205/cma.1698981.
JAMA Chesneau C. Study of some new one-parameter modifications of the Hardy-Hilbert integral inequality. CMA. 2025;8:93–116.
MLA Chesneau, Christophe. “Study of Some New One-Parameter Modifications of the Hardy-Hilbert Integral Inequality”. Constructive Mathematical Analysis, vol. 8, no. 2, 2025, pp. 93-116, doi:10.33205/cma.1698981.
Vancouver Chesneau C. Study of some new one-parameter modifications of the Hardy-Hilbert integral inequality. CMA. 2025;8(2):93-116.