New integral formulas with applications to integral inequalities

Christophe Chesneau

doi:10.2478/awutm-2025-0008

Abstract

In this article, we derive new integral formulas involving a ratio function, a maximum function, and three adjustable parameters. Two of these parameters control the maximum function in di erent ways. The arctangent function plays a central role in the resulting expressions. These formulas are then used to construct new and varied types of integral inequalities. In particular, we present weighted Hölder-type integral inequalities, as well as new Hardy-Hilbert-type integral inequalities. Their novelty lies mainly in the inclusion of the maximum function and the two parameters governing it. Detailed proofs are given.

References

L. E. Azar, On some extensions of Hardy-Hilbert’s inequality and applications, J. Ineq. Appl. 2008 (2008), 1-14.
Search in Google Scholar Back to article
D. Bainov, P. Simeonov, Integral Inequalities and Applications, Kluwer Academic, Dordrecht, 1992.
Search in Google Scholar Back to article
E. F. Beckenbach, R. Bellman, Inequalities, Springer, Berlin, 1961.
Search in Google Scholar Back to article
Q. Chen, B. C. Yang, A survey on the study of Hilbert-type inequalities, J. Inequal. Appl. 2015 (2015), 1-29.
Search in Google Scholar Back to article
Z. Cvetkovski, Inequalities: Theorems, Techniques and Selected Problems, SpringerLink : Bücher, Springer Berlin Heidelberg, 2012.
Search in Google Scholar Back to article
I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 7th Edition, Academic Press, 2007.
Search in Google Scholar Back to article
G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1934.
Search in Google Scholar Back to article
Y. Li, J. Wu, B. He, A new Hilbert-type integral inequality and the equivalent form, Int. J. Math. Math. Sci. 8 (2006), 1-6.
Search in Google Scholar Back to article
A. Saglam, H. Yildirim, M. Z. Sarikaya, Generalization of Hardy-Hilbert’s inequality and applications, Kyungpook Math. J. 50 (2010), 131-152.
Search in Google Scholar Back to article
M. Z. Sarikaya, M. S. Bingol, Recent developments of integral inequalities of the Hardy-Hilbert type, Turkish J. Ineq. 8 (2024), 43-54.
Search in Google Scholar Back to article
W. T. Sulaiman, New Hardy-Hilbert’s-type integral inequalities, Int. Math. Forum 3 (2008), 2139-2147.
Search in Google Scholar Back to article
W. T. Sulaiman, New kinds of Hardy-Hilbert’s integral inequalities, Appl. Math. Lett. 23 (2010), 361-365.
Search in Google Scholar Back to article
W. T. Sulaiman, An extension of Hardy-Hilbert’s integral inequality, Afr. Diaspora J. Math. 10 (2010), 66-71.
Search in Google Scholar Back to article
B. Sun, Best generalization of a Hilbert type inequality, J. Ineq. Pure Appl. Math. 7 (2006), 1-7.
Search in Google Scholar Back to article
W. Walter, Di erential and Integral Inequalities, Springer, Berlin, 1970.
Search in Google Scholar Back to article
B. C. Yang, Hilbert-Type Integral Inequalities, Bentham Science Publishers, The United Arab Emirates, 2009.
Search in Google Scholar Back to article
B. C. Yang, The Norm of Operator and Hilbert-Type Inequalities, Science Press, Beijing, 2009.
Search in Google Scholar Back to article

New integral formulas with applications to integral inequalities

Abstract

Paradigm

My account