Some New Young-type Integral Inequalities
Abstract
In this article, we present new forms of Young-type integral inequalities. Some of these extend the classical Young integral inequality by including a function and its inverse, while others introduce a function and its ratio transformation or derivative. The proofs of the main results are based on the standard Young product inequality and well-calibrated analytical tools, such as the Chasles integral formula and the Cauchy-Schwarz integral inequality, as well as changes of variables. Several inequalities also address the case where the main function is decreasing, thus expanding the traditional framework, which assumes an increasing function. In particular, the decreasing case leads to elegant formulations. Some theoretical aspects are illustrated with examples.
Downloads
References
Hardy, G. H., Littlewood, J. E., & Polya, G. (1934). Inequalities. Cambridge University Press.
Beckenbach, E. F., & Bellman, R. (1961). Inequalities. Springer-Verlag.
Walter, W. (1970). Differential and integral inequalities (Die Grundlehren der mathematischen Wissenschaften, Band 164). Springer-Verlag. https://doi.org/10.1007/978-3-642-86405-6
Bainov, D., & Simeonov, P. (1992). Integral inequalities and applications (Mathematics and Its Applications, Vol. 57). Kluwer Academic Publishers.
Cvetkovski, Z. (2012). Inequalities: Theorems, techniques and selected problems. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-23792-8
Young, W. H. (1912). On classes of summable functions and their Fourier series. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 87(594), 225-229. https://doi.org/10.1098/rspa.1912.0076
Anderson, D. R. (2007). Young's integral inequality on time scales revisited. Journal of Inequalities in Pure and Applied Mathematics, 8(3), Article 64.
Boas, R. P., Jr., & Marcus, M. B. (1974). Generalizations of Young's inequality. Journal of Mathematical Analysis and Applications, 46(1), 36-40. https://doi.org/10.1016/0022-247X(74)90279-0
Boas, R. P., Jr., & Marcus, M. B. (1973). Inequalities involving a function and its inverse. SIAM Journal on Mathematical Analysis, 4(4), 585-591. https://doi.org/10.1137/0504051
Cerone, P. (2009). On Young's inequality and its reverse for bounding the Lorenz curve and Gini mean. Journal of Mathematical Inequalities, 3(3), 369-381. https://doi.org/10.7153/jmi-03-37
Chesneau, C. (2025). On a generalization of the Young inequality involving intermediary functions and its applications. Mathematica Pannonica, 30(2), 223-243.
Chesneau, C. (2025). New results on composed variants of the Young integral inequality. Surveys in Mathematics and its Applications, 20, 75-106.
Cooper, R. (1927). Notes on certain inequalities: (I); Generalization of an inequality of W. H. Young. Journal of the London Mathematical Society, s1-2(1), 17-21. https://doi.org/10.1112/jlms/s1-2.1.17
Cooper, R. (1927). Notes on certain inequalities: II. Journal of the London Mathematical Society, s1-2(3), 159-163. https://doi.org/10.1112/jlms/s1-2.3.159
Cunningham, F., Jr., & Grossman, N. (1971). On Young's inequality. The American Mathematical Monthly, 78(7), 781-783. https://doi.org/10.2307/2318018
Diaz, J. B., & Metcalf, F. T. (1970). An analytic proof of Young's inequality. The American Mathematical Monthly, 77(6), 603-609. https://doi.org/10.2307/2316736
Hsu, I. C. (1972). On a converse of Young's inequality. Proceedings of the American Mathematical Society, 33(1), 107-108. https://doi.org/10.2307/2038179
Lackovic, I. A. (1974). A note on a converse of Young's inequality. Univerzitet u Beogradu. Publikacije Elektrotehnickog Fakulteta. Serija Matematika i Fizika, (461-497), 73-76.
Mitrinovic, D. S. (1970). Analytic inequalities (Die Grundlehren der mathematischen Wissenschaften, Band 165). Springer-Verlag.
Mitrinovic, D. S., Pecaric, J. E., & Fink, A. M. (1993). Classical and new inequalities in analysis (Mathematics and Its Applications, Vol. 61). Kluwer Academic Publishers. https://doi.org/10.1007/978-94-017-1043-5
Mitroi, F.-C., & Niculescu, C. P. (2011). An extension of Young's inequality. Abstract and Applied Analysis, 2011, Article 162049. https://doi.org/10.1155/2011/162049
Oppenheim, A. (1927). Note on Mr. Cooper's generalization of Young's inequality. Journal of the London Mathematical Society, s1-2(1), 21-23. https://doi.org/10.1112/jlms/s1-2.1.21
Pales, Z. (1992). A general version of Young's inequality. Archiv der Mathematik, 58(4), 360-365. https://doi.org/10.1007/BF01189925
Pales, Z. (1987). A generalization of Young's inequality. In W. Walter (Ed.), General inequalities 5 (International Series of Numerical Mathematics, Vol. 80, pp. 471-472). Birkhauser.
Pales, Z. (1990). On Young-type inequalities. Acta Scientiarum Mathematicarum, 54(3-4), 327-338.
Parker, F. D. (1955). Integrals of inverse functions. The American Mathematical Monthly, 62(6), 439-440. https://doi.org/10.2307/2307006
Ruthing, D. (1994). On Young's inequality. International Journal of Mathematical Education in Science and Technology, 25(2), 161-164. https://doi.org/10.1080/0020739940250201
Sulaiman, W. T. (2009). Notes on Young's inequality. International Mathematical Forum, 4(24), 1173-1180.
Witkowski, A. (2007). On Young's inequality. Journal of Inequalities in Pure and Applied Mathematics, 8(5), Article 164.
Wong, F.-H., Yeh, C.-C., Yu, S.-L., & Hong, C.-H. (2005). Young's inequality and related results on time scales. Applied Mathematics Letters, 18(9), 983-988. https://doi.org/10.1016/j.aml.2004.06.028
Zhu, L. (2004). On Young's inequality. International Journal of Mathematical Education in Science and Technology, 35(4), 601-603. https://doi.org/10.1080/00207390410001686698
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
