Two Alternative Proofs of the Grüss Inequality
DOI:
https://doi.org/10.21467/ias.10.1.78-84Abstract
The classical Grüs inequality has spurred a range of improvements, generalizations, and extensions. In this article, we provide new functional bounds that ultimately lead to two elementary proofs of the inequality that might be of interest. Our results are motivated by the extreme cases where the equality is reached, namely step functions of equal support. Our first proof is based on the standard Cauchy-Schwarz inequality and a simple bound on the variance of a function. Its simplicity would be of particular interest to those who are new to the study of functional inequalities. Our second proof utilizes non-intuitive and novel bounds on functionals defined on L∞(0, 1). As a result, we provide a detailed and new insight into the nature of the Grüss inequality.
Keywords:
Inequalities, Gruss inequality, Functional inequalitiesDownloads
References
Mercer, A M.: An improvement of the Grüss inequality. In: JIPAM. J. Inequal. Pure Appl. Math 6 (2005), Nr. 4, S. 93
Dragomir, Sever S.: Some Grüss-type results via Pompeiu’s-like inequalities. In: Arabian Journal of Mathematics 4 (2015), Nr. 3, 159–170. https://doi.org/10.1007/s40065-015-0135-8
Niezgoda, Marek: A new inequality of Ostrowski–Grüss type and applications to some numerical quadrature rules. In: Computers & Mathematics with Applications 58 (2009), Nr. 3, 589–596. https://doi.org/10.1016/j.camwa.2009.03.089
Farid, Ghulam u. a.: Straightforward proofs of Ostrowski inequality and some related results. In: Int. J. Ana 5 (2016). https://doi.org/10.1155/2016/3918483
Gruss, Gerhard: Über dasMaximum des absoluten Betrages von. In: Mathematische Zeitschrift 39 (1935), Nr. 1, 215–226. https://doi.org/10.1007/BF01201355
Mercer, A M. ; Mercer, PR: New proofs of the Grüss inequality. In: Aust. J. Math. Anal. Appl 1 (2004), Nr. 2, S. 6
Mitrinovic, Dragoslav S. ; Pecaric, Josip ; Fink, Arlington M.: Classical and new inequalities in analysis. Bd. 61. Springer Science & Business Media, 2013. https://doi.org/10.1007/978-94-017-1043-5
Downloads
Published
Issue
Section
How to Cite
License
Copyright (c) 2020 Martin Tchernookov
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Author(s) retains full copyright of their article and grants non-exclusive publishing right to International Annals of Science and its publisher “AIJR (India)”. Author(s) can archive pre-print, post-print and published version/PDF to any open access, institutional repository, social media or personal website provided that Published source must be acknowledged with citation and link to publisher version.
Click here for more information on Copyright policy
Click here for more information on Licensing policy
