Two new theorems on integral inequalities involving maximums and minimums of ratios

Christophe Chesneau

doi:10.14244/lajm.v5i1.64

Authors

Christophe Chesneau University of Caen-Normandie https://orcid.org/0000-0002-1522-9292

DOI:

https://doi.org/10.14244/lajm.v5i1.64

Keywords:

Two-dimensional integral inequalities, Hardy-Hilbert-type integral inequalities, beta function, gamma function

Abstract

In this article, we present two new theorems relating to the upper bounds of specific two-dimensional integral inequalities. The first theorem uses certain maximums of the ratios of the two variables, while the second offers an analogous result based on certain minimums of these ratios. The obtained bounds are quite manageable, with constant factors defined by simple integrals. Complete proofs are provided for the sake of rigor. To illustrate the scope and implications of these results, we present several examples, some of which focus on the standard beta and gamma functions.

Author Biography

Christophe Chesneau, University of Caen-Normandie

Department of Mathematics, LMNO

References

[1] L. E. Azar, On some extensions of Hardy-Hilbert’s inequality and applications, J. Inequal. Appl. 2008 (2008), 1–14. DOI: https://doi.org/10.1155/2008/546829

[2] Q. Chen and B. C. Yang, A survey on the study of Hilbert-type inequalities, J. Inequal. Appl. 2015 (2015), 1–29. DOI: https://doi.org/10.1186/s13660-015-0829-7

[3] C. Chesneau, Refining and extending two special Hardy-Hilbert-type integral inequalities, Ann. Math. Comput. Sci. 28 (2025), 21–45. DOI: https://doi.org/10.56947/amcs.v28.513

[4] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1934.

[5] Y. Li, J. Wu, and B. He, A new Hilbert-type integral inequality and the equivalent form, Int. J. Math. Math. Sci. 8 (2006), 1–6. DOI: https://doi.org/10.1155/IJMMS/2006/45378

[6] A. Saglam, H. Yildirim, and M. Z. Sarikaya, Generalization of Hardy-Hilbert’s inequality and applications, Kyungpook Math. J. 50 (2010), 131–152. DOI: https://doi.org/10.5666/KMJ.2010.50.1.131

[7] M. Z. Sarikaya and M. S. Bingol, Recent developments of integral inequalities of the Hardy-Hilbert type, Turkish J. Inequal. 8 (2024), 43–54.

[8] W. T. Sulaiman, New Hardy-Hilbert’s-type integral inequalities, Int. Math. Forum 3 (2008), 2139–2147.

[9] W. T. Sulaiman, Hardy-Hilbert’s integral inequality in new kinds, Math. Commun. 15 (2010), 453–461.

[10] W. T. Sulaiman, New kinds of Hardy-Hilbert’s integral inequalities, Appl. Math. Lett. 23 (2010), 361–365. DOI: https://doi.org/10.1016/j.aml.2009.10.011

[11] W. T. Sulaiman, An extension of Hardy-Hilbert’s integral inequality, Afr. Diaspora J. Math. 10 (2010), 66–71.

[12] B. Sun, Best generalization of a Hilbert type inequality, J. Inequal. Pure Appl. Math. 7 (2006), 1–7. DOI: https://doi.org/10.1155/2007/71049

[13] B. C. Yang, Hilbert-Type Integral Inequalities, Bentham Science Publishers, The United Arab Emirates, 2009.

[14] B. C. Yang, The Norm of Operator and Hilbert-Type Inequalities,Science Press, Beĳing, 2009. DOI: https://doi.org/10.2174/97816080505501090101