Li, Xin; Mohapatra, E. N.; Rodriguez, R. S. Grüss-type inequalities. (English) Zbl 1007.26016 J. Math. Anal. Appl. 267, No. 2, 434-443 (2002). This paper discusses the connection between the Grüss inequality and the best approximation of functions by constants. Let \((X,\Sigma,\mu)\) be a probability space and let \(f,g\in L^{2}(\mu).\) Then \[ \left|\int_{X}fg d\mu-\int _{X}f d\mu\cdot\int_{X}g d\mu\right|\leq\inf\|f-\alpha\|_{L^{2}}\cdot\inf\|g-\beta\|_{L^{2}} \] and this fact is used to recover the discrete form of the Grüss inequality, as formulated by Biernacki, Pidek and Ryll-Nardzewski. Also, its connection with the original result of Grüss is presented in great detail. In the same spirit, Theorem 10 gives another Grüss-type inequality for the functions \(f\in L^{\infty}(\mu)\), which are orthogonal on all polynomials of degree \(\leq n:\) \[ \int_{X}|f|d\mu \leq\inf\{ \|f-P\|_{L^{\infty}} ;\text{ }P\text{ polynomial of degree }\leq n\}. \] Reviewer: Constantin Niculescu (Craiova) Cited in 26 Documents MSC: 26D15 Inequalities for sums, series and integrals 41A50 Best approximation, Chebyshev systems Keywords:Grüss inequality; best approximation PDFBibTeX XMLCite \textit{X. Li} et al., J. Math. Anal. Appl. 267, No. 2, 434--443 (2002; Zbl 1007.26016) Full Text: DOI References: [1] Dragomir, S. S.; Fedotov, I., An inequality of Grüss type for Riemann-Stieljes integral and applications for special means, Tamkang J. Math., 29, 287-292 (1998) · Zbl 0924.26013 [2] Dragomir, S. S., A generalization of Grüss inequality in inner product spaces and applications, J. Math. Anal. Appl., 237, 74-82 (1999) · Zbl 0943.46011 [3] Dragomir, S. S.; Wang, S., An inequality of Ostrowski-Grüss type and its applications to the estimation of error bounds for some means and for some numerical quadrature rules, Comput. Math. Appl., 33, 15-20 (1997) · Zbl 0880.41025 [4] Fedotov, I.; Dragomir, S. S., An inequality of Ostrowski type and its applications for Simpson’s rule and special means, Math. Inequal. Appl., 2, 491-499 (1999) · Zbl 0944.26026 [5] Fink, A. M., A treatise on Grüss’ inequality, (Rassias, T. M.; Srivatava, H. M., Analytic and Geomatric Inequalities and Applications (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 93-113 · Zbl 0982.26012 [6] Grüss, G., Uber das maximum des absoluten betrages von \(1b−a∫a^bf(x)g(x)dx−1(b−a)^2\)∫\(a^b\) f(x)dx∫\(a^b\) g(x)dx\), Math. Z., 39, 215-226 (1935) · JFM 60.0189.02 [7] R. Jones, X. Li, R. N. Mohapatra, and, R. S. Rodriguez, An elementary proof of Grüss inequality, IAMG Report, 1, 2000.; R. Jones, X. Li, R. N. Mohapatra, and, R. S. Rodriguez, An elementary proof of Grüss inequality, IAMG Report, 1, 2000. [8] Mitrinovic, D. S.; Pecaric, J. E.; Fink, A. M., Classical and New Inequalities in Analysis (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0771.26009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.