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Year 2025, Volume: 8 Issue: 2, 100 - 116, 01.07.2025

Christophe Chesneau

https://doi.org/10.33434/cams.1673361

Abstract

References

  • [1] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1934.
  • [2] E. F. Beckenbach, R. Bellman, Inequalities, Springer, Berlin, 1961.
  • [3] W. Walter, Differential and Integral Inequalities, Springer, Berlin, 1970.
  • [4] D. Bainov, P. Simeonov, Integral Inequalities and Applications, Mathematics and Its Applications, Vol. 57. Kluwer Academic, Dordrecht, 1992.
  • [5] B. C. Yang, Hilbert-Type Integral Inequalities, Bentham Science Publishers, The United Arab Emirates, 2009.
  • [6] 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 (2013), 391-401.
  • [7] V. Adiyasuren, T. Batbold, M. Krnić, Hilbert-type inequalities involving differential operators, the best constants and applications, Math. Inequal. Appl., 18 (2015), 111-124. https://doi.org/10.7153/mia-18-07
  • [8] C. Chesneau, Some four-parameter trigonometric generalizations of the Hilbert integral inequality, Asia Math., 8 (2024), 45-59. https://doi.org/10.5281/zenodo.13949386
  • [9] C. Chesneau, Study of two three-parameter non-homogeneous variants of the Hilbert integral inequality, Lobachevskii J. Math., 45 (2024), 4931-4953. https://doi.org/10.1134/S1995080224605526
  • [10] Q. Chen, B. C. Yang, A survey on the study of Hilbert-type inequalities, J. Inequal. Appl., 2015 (2015), 1-29. https://doi.org/10.1186/s13660-015-0829-7
  • [11] W. T. Sulaiman, New types of Hardy-Hilbert’s integral inequality, Gen. Math. Notes, 2 (2011), 111-118.
  • [12] W. W. Wei, X. Y. Lei, Some more generalizations of the integral inequalities of Hardy and Hilbert, Hacet. J. Math. Stat., 40 (2011), 863-869.
  • [13] Z. Huang, B. C. Yang, A multidimensional Hilbert-type integral inequality, J. Inequal. Appl., 2015 (2015), Article ID 151, 13 pages. https://doi.org/10.1186/s13660-015-0673-9
  • [14] M. Z. Sarikaya, M. S. Bingol, Recent developments of integral inequalities of the Hardy-Hilbert type, Turkish J. Ineq., 8(2) (2024), 43-54.
  • [15] J. F. Tian, Properties of generalized H¨older’s inequalities, J. Math. Ineq., 9(2) (2015), 473-480. https://doi.org/10.7153/jmi-09-40
  • [16] B. Benaissa, M. Z. Sarikaya, On the refinements of some important inequalities with a finite set of positive numbers, Math.Methods Appl. Sci., 47(12) (2024), 9589-9599. https://doi.org/10.1002/mma.10084
  • [17] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 7th Edition, Academic Press, Amsterdam, 2007.

General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology

Year 2025, Volume: 8 Issue: 2, 100 - 116, 01.07.2025

Christophe Chesneau

https://doi.org/10.33434/cams.1673361

Abstract

This article establishes new general two- and three-dimensional integral inequalities. The first result involves four functions: two main functions defined on the positive real line and two auxiliary functions defined on the unit interval. As a significant contribution, the upper bound obtained is quite simple; it is expressed only as the product of the unweighted integral norms of these functions. The main ingredient of the proof is an original change of variables methodology. The article also presents a three-dimensional extension of this result. This higher-dimensional version uses a similar structure but with nine functions: three main functions defined on the positive real line and six auxiliary functions defined on the unit interval. It retains the simplicity and sharpness of the upper bound. Both results open up new directions for applications in analysis. This claim is supported by various examples, including some based on power, logarithmic, trigonometric, and exponential functions, as well as some secondary but still general integral inequalities.

Keywords

Change of variables Gamma function Hardy-Hilbert-type integral inequalities Three-dimensional integral inequalities Two-dimensional integral inequalities

References

  • [1] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1934.
  • [2] E. F. Beckenbach, R. Bellman, Inequalities, Springer, Berlin, 1961.
  • [3] W. Walter, Differential and Integral Inequalities, Springer, Berlin, 1970.
  • [4] D. Bainov, P. Simeonov, Integral Inequalities and Applications, Mathematics and Its Applications, Vol. 57. Kluwer Academic, Dordrecht, 1992.
  • [5] B. C. Yang, Hilbert-Type Integral Inequalities, Bentham Science Publishers, The United Arab Emirates, 2009.
  • [6] 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 (2013), 391-401.
  • [7] V. Adiyasuren, T. Batbold, M. Krnić, Hilbert-type inequalities involving differential operators, the best constants and applications, Math. Inequal. Appl., 18 (2015), 111-124. https://doi.org/10.7153/mia-18-07
  • [8] C. Chesneau, Some four-parameter trigonometric generalizations of the Hilbert integral inequality, Asia Math., 8 (2024), 45-59. https://doi.org/10.5281/zenodo.13949386
  • [9] C. Chesneau, Study of two three-parameter non-homogeneous variants of the Hilbert integral inequality, Lobachevskii J. Math., 45 (2024), 4931-4953. https://doi.org/10.1134/S1995080224605526
  • [10] Q. Chen, B. C. Yang, A survey on the study of Hilbert-type inequalities, J. Inequal. Appl., 2015 (2015), 1-29. https://doi.org/10.1186/s13660-015-0829-7
  • [11] W. T. Sulaiman, New types of Hardy-Hilbert’s integral inequality, Gen. Math. Notes, 2 (2011), 111-118.
  • [12] W. W. Wei, X. Y. Lei, Some more generalizations of the integral inequalities of Hardy and Hilbert, Hacet. J. Math. Stat., 40 (2011), 863-869.
  • [13] Z. Huang, B. C. Yang, A multidimensional Hilbert-type integral inequality, J. Inequal. Appl., 2015 (2015), Article ID 151, 13 pages. https://doi.org/10.1186/s13660-015-0673-9
  • [14] M. Z. Sarikaya, M. S. Bingol, Recent developments of integral inequalities of the Hardy-Hilbert type, Turkish J. Ineq., 8(2) (2024), 43-54.
  • [15] J. F. Tian, Properties of generalized H¨older’s inequalities, J. Math. Ineq., 9(2) (2015), 473-480. https://doi.org/10.7153/jmi-09-40
  • [16] B. Benaissa, M. Z. Sarikaya, On the refinements of some important inequalities with a finite set of positive numbers, Math.Methods Appl. Sci., 47(12) (2024), 9589-9599. https://doi.org/10.1002/mma.10084
  • [17] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 7th Edition, Academic Press, Amsterdam, 2007.
There are 17 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Articles
Authors

Christophe Chesneau 0000-0002-1522-9292

Early Pub Date June 28, 2025
Publication Date July 1, 2025
Submission Date April 10, 2025
Acceptance Date June 27, 2025
Published in Issue Year 2025 Volume: 8 Issue: 2

Cite

APA Chesneau, C. (2025). General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology. Communications in Advanced Mathematical Sciences, 8(2), 100-116. https://doi.org/10.33434/cams.1673361
AMA Chesneau C. General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology. Communications in Advanced Mathematical Sciences. July 2025;8(2):100-116. doi:10.33434/cams.1673361
Chicago Chesneau, Christophe. “General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology”. Communications in Advanced Mathematical Sciences 8, no. 2 (July 2025): 100-116. https://doi.org/10.33434/cams.1673361.
EndNote Chesneau C (July 1, 2025) General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology. Communications in Advanced Mathematical Sciences 8 2 100–116.
IEEE C. Chesneau, “General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology”, Communications in Advanced Mathematical Sciences, vol. 8, no. 2, pp. 100–116, 2025, doi: 10.33434/cams.1673361.
ISNAD Chesneau, Christophe. “General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology”. Communications in Advanced Mathematical Sciences 8/2 (July 2025), 100-116. https://doi.org/10.33434/cams.1673361.
JAMA Chesneau C. General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology. Communications in Advanced Mathematical Sciences. 2025;8:100–116.
MLA Chesneau, Christophe. “General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology”. Communications in Advanced Mathematical Sciences, vol. 8, no. 2, 2025, pp. 100-16, doi:10.33434/cams.1673361.
Vancouver Chesneau C. General Two- and Three-Dimensional Integral Inequalities Based a Change of Variables Methodology. Communications in Advanced Mathematical Sciences. 2025;8(2):100-16.
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