In its standard form, Young's inequality for products is a mathematical inequality about the product of two numbers and it allows us to estimate a product of two terms by a sum of the same terms raised to a power and scaled. This inequality, though very simple, has attracted researchers working in many fields of mathematics due to its applications. Apart from the above standard form, there are numerous refinements and variants of Young’s inequality in the literature. Some of these variants are Young’s inequality for arbitrary products, Young’s inequality for increasing functions, Young’s inequality for convolutions, Young’s inequality for integrals, Young’s inequality for matrices, trace version of Young’s inequality, determinant version of Young’s inequality, and so on. The present study examines three variants of Young’s inequality, namely the standard Young’s inequality, Young’s inequality for increasing functions and Young’s inequality for arbitrary products. There are various proofs for these three variants in the literature. For example, just like several other classical important inequalities, these inequalities can be deduced from Jensen’s inequality. The objective of this article is to provide a new alternative proof for each of them. The significance of the article lies in its attempts to open a new direction of poof so that the same approach could be applied to other useful inequalities. The proofs to be presented are based on the methods of multivariable optimization theory.
| Published in | Mathematics Letters (Volume 7, Issue 4) |
| DOI | 10.11648/j.ml.20210704.12 |
| Page(s) | 54-58 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Young’s Inequality, Multivariable Extrema, Convolution, Cauchy-Schwarz Inequality, Holder’s Inequality, Jensen Inequality
| [1] | M. Akkouchi: Cauchy-Schwarz inequality implies Holder’s inequality, RGMIA Research report collection, vol. 21, Art. ID. 48, 2018. |
| [2] | T. Ando: Matrix Young inequality, Oper. Theory Adv. Appl. 75 (1995) 33–38. |
| [3] | P. K. Bhandari and S. K. Bissu: Inequalities via Holder’s inequality, Scholars journal of research in mathematics and computer science, vol. 2, no. 2, pp. 124-129, 2018. |
| [4] | P. S. Bullen: A dictionary of inequalities, Pitman Monographs and Surveys in Pure and applied mathematics 97, Addison Wesley Longman Ltd. UK., 1998. |
| [5] | P. S. Bullen: Handbook of Means and Their Inequalities, Kluwer Academic Publishers, Dordrecht, 2003. doi: 10.1007/978-94-017-0399-4. |
| [6] | G. H. Hardy, J. E. Littlewood and G. Polya: Inequalities, Cambridge, 1934. |
| [7] | Herman, R. Kucera, J. Simsa: Equations and inequalities, CMS Books in Mathematics, Springer 2000. |
| [8] | C. A. Infantozzi: An introduction to relations among inequalities, The November Meeting in Cleveland, Ohio November 25, 1972, Notices of the American mathematical society, no. 141, pp. A819-A820, 1972. |
| [9] | Y. C. Li and S. Y. Shaw: A proof of Hölder’s inequality using the Cauchy-Schwarz inequality, Journal of inequalities in pure and applied mathematics, vol. 7, no. 2, Art. ID. 62, 2006. |
| [10] | A. Lohwater: Introduction to Inequalities. e-book in PDF. http://www.mediafire.com/?1mw1tkgozzu, 1982. |
| [11] | D. S. Mitrinovic: Analytic Inequalities, Springer-Verlag, Berlin Heiidelberg 1970. |
| [12] | D. S. Mitrinovic, J. E. Pečarić and A. M. Fink: Classical and new inequalities in analysis, Dordrecht: Kluwer Academic Publishers, 1993. |
| [13] | F. C. Mitroi & C. P. Niculescu: An extension of Young's inequality in Abstract and Applied Analysis (Vol. 2011). Hindawi. |
| [14] | Z. Pales: On Young-type inequalities, Acta Scientiarum Mathematicarum, vol. 54, no. 3-4, pp. 327–338, 1990. |
| [15] | J. M. Steel: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities, Cambridge University Press, New York, 2004. |
| [16] | W. T. Sulaiman: Notes on Youngs's inequality, International Mathematical Forum, vol. 4, no. 21–24, pp. 1173–1180, 2009. |
| [17] | E. Tolsted: An elementary derivation of the Cauchy, Holder, and Minkowski inequality- ties from Young's inequality, Math. Mag., 37 (1964) 2-12. |
| [18] | W. H. Young: On classes of summable functions and their Fourier series, Proc. Royal Soc., Series (A), 87 (1912) 225-229. |
| [19] | J. Zhang, J. Wu: New progress on the operator inequalities involving improved Young’s and its reverse inequalities relating to the Kantorovich constant. J Inequal Appl 2017, 69 (2017). |
APA Style
Tadesse Bekeshie. (2022). A New Proof of Young’s Inequality Using Multivariable Optimization. Mathematics Letters, 7(4), 54-58. https://doi.org/10.11648/j.ml.20210704.12
ACS Style
Tadesse Bekeshie. A New Proof of Young’s Inequality Using Multivariable Optimization. Math. Lett. 2022, 7(4), 54-58. doi: 10.11648/j.ml.20210704.12
AMA Style
Tadesse Bekeshie. A New Proof of Young’s Inequality Using Multivariable Optimization. Math Lett. 2022;7(4):54-58. doi: 10.11648/j.ml.20210704.12
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ml.20210704.12,
author = {Tadesse Bekeshie},
title = {A New Proof of Young’s Inequality Using Multivariable Optimization},
journal = {Mathematics Letters},
volume = {7},
number = {4},
pages = {54-58},
doi = {10.11648/j.ml.20210704.12},
url = {https://doi.org/10.11648/j.ml.20210704.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20210704.12},
abstract = {In its standard form, Young's inequality for products is a mathematical inequality about the product of two numbers and it allows us to estimate a product of two terms by a sum of the same terms raised to a power and scaled. This inequality, though very simple, has attracted researchers working in many fields of mathematics due to its applications. Apart from the above standard form, there are numerous refinements and variants of Young’s inequality in the literature. Some of these variants are Young’s inequality for arbitrary products, Young’s inequality for increasing functions, Young’s inequality for convolutions, Young’s inequality for integrals, Young’s inequality for matrices, trace version of Young’s inequality, determinant version of Young’s inequality, and so on. The present study examines three variants of Young’s inequality, namely the standard Young’s inequality, Young’s inequality for increasing functions and Young’s inequality for arbitrary products. There are various proofs for these three variants in the literature. For example, just like several other classical important inequalities, these inequalities can be deduced from Jensen’s inequality. The objective of this article is to provide a new alternative proof for each of them. The significance of the article lies in its attempts to open a new direction of poof so that the same approach could be applied to other useful inequalities. The proofs to be presented are based on the methods of multivariable optimization theory.},
year = {2022}
}
TY - JOUR T1 - A New Proof of Young’s Inequality Using Multivariable Optimization AU - Tadesse Bekeshie Y1 - 2022/01/24 PY - 2022 N1 - https://doi.org/10.11648/j.ml.20210704.12 DO - 10.11648/j.ml.20210704.12 T2 - Mathematics Letters JF - Mathematics Letters JO - Mathematics Letters SP - 54 EP - 58 PB - Science Publishing Group SN - 2575-5056 UR - https://doi.org/10.11648/j.ml.20210704.12 AB - In its standard form, Young's inequality for products is a mathematical inequality about the product of two numbers and it allows us to estimate a product of two terms by a sum of the same terms raised to a power and scaled. This inequality, though very simple, has attracted researchers working in many fields of mathematics due to its applications. Apart from the above standard form, there are numerous refinements and variants of Young’s inequality in the literature. Some of these variants are Young’s inequality for arbitrary products, Young’s inequality for increasing functions, Young’s inequality for convolutions, Young’s inequality for integrals, Young’s inequality for matrices, trace version of Young’s inequality, determinant version of Young’s inequality, and so on. The present study examines three variants of Young’s inequality, namely the standard Young’s inequality, Young’s inequality for increasing functions and Young’s inequality for arbitrary products. There are various proofs for these three variants in the literature. For example, just like several other classical important inequalities, these inequalities can be deduced from Jensen’s inequality. The objective of this article is to provide a new alternative proof for each of them. The significance of the article lies in its attempts to open a new direction of poof so that the same approach could be applied to other useful inequalities. The proofs to be presented are based on the methods of multivariable optimization theory. VL - 7 IS - 4 ER -
Email: