We discuss the Grüss inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain the Grüss inequality for the functional L(f) = H(f; x), where H:C[a, b] → C[a, b] is a positive linear operator and x ∈ [a, b] is fixed. We apply this inequality in the case of known operators, e.g., the Bernstein operator, the Hermite–Fejér interpolation operator, and convolution-type operators. Moreover, we deduce Grüss-type inequalities using the Cauchy mean-value theorem, thus generalizing results of Chebyshev and Ostrowski. The Grüss inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained. The latter, in turn, leads to one further version of the Grüss inequality. In the appendix, we prove a new result concerning the absolute first-order moments of the classic Hermite–Fejér operator.
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References
A. M. Acu and H. Gonska, “Ostrowski-type inequalities and moduli of smoothness,” Results Math., 53, 217–228 (2009).
D. Andrica and C. Badea, “Grüss inequality for positive linear functionals,” Period. Math. Hung., 19, 155–167 (1988).
G. A. Anastassiou, “Ostrowski type inequalities,” Proc. Amer. Math. Soc., 123, 3775–3781 (1995).
P. L. Chebyshev, “Sur les expressions approximatives des intégrales définies par les autres prises entre les mêmes limites,” Proc. Math. Soc. Kharkov, 2, 93–98 (1882); French translation: Oeuvres, 2, 716–719 (1907).
X. L. Cheng, “Improvement of some Ostrowski–Grüss type inequalities,” Comput. Math. Appl., 42, 109–114 (2001).
S. S. Dragomir, “On the Ostrowski integral inequality for Lipschitzian mappings and applications,” Comput. Math. Appl., 38, 33–37 (1999).
S. S. Dragomir and S. Wang, “An inequality of Ostrowski–Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules,” Comput. Math. Appl., 33, No. 11, 15–20 (1997).
B. Gavrea and I. Gavrea, “Ostrowski type inequalities from a linear functional point of view,” J. Inequal. Pure Appl. Math., 1, Article 11 (2000).
H. Gonska, “Quantitative Approximation in C(X),” Habilitationsschrift, University of Duisburg (1986).
S. J. Goodenough and T. M. Mills, “A new estimate for the approximation of functions by Hermite–Fejér interpolation polynomials,” J. Approxim. Theory, 31, 253–260 (1981).
G. Grüss, “Über das Maximum des absoluten Betrages von \( \frac{1}{{b - a}}\int_a^b {f(x)g(x)dx - \frac{1}{{{{\left( {b - a} \right)}^2}}}} \int_a^b {f(x)dx} \int_a^b {g(x)dx} \),” Math. Z., 39, 215–226 (1935).
J. Karamata, “Inégalités relatives aux quotients et à la différence de \( \int {fg\;{\text{et}}\;} \int f \int g \),” Bull. Acad. Serbe. Sci. Math. Natur. A, 131–145 (1948).
O. Kiš, “Remarks on the rapidity of convergence of Lagrange interpolation,” Ann. Univ. Sci. Budapest. Sec. Math., 11, 27–40 (1968).
E. Landau, “Über einige Ungleichungen von Herrn G. Grüss,” Math. Z., 39, 742–744 (1935).
H. G. Lehnhoff, “A simple proof of A. F. Timan’s theorem,” J. Approxim. Theory, 38, 172–176 (1983).
Y. Matsuoka, “On the degree of approximation of functions by some positive linear operators,” Sci. Rep. Kagoshima Univ., 9, 11–16 (1960).
A. Mc. D. Mercer and P. R. Mercer, “New proofs of the Grüss inequality,” Austral. J. Math. Anal. Appl., 1, Issue 2, 1–6 (2004).
R. N. Misra, “On the rate of convergence of Hermite–Fejér interpolation polynomials,” Period. Math. Hung., 13, 15–20 (1982).
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer, Dordrecht (1993).
A. Ostrowski, “Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert,” Comment. Math. Helv., 10, 226–227 (1938).
A. Ostrowski, “On an integral inequality,” Aequat. Math., 4, 358–373 (1970).
B. G. Pachpatte, “A note on Ostrowski like inequalities,” J. Inequal. Pure Appl. Math., 6, Article 114 (2005).
B. G. Pachpatte, “A note on Grüss type inequalities via Cauchy’s mean-value theorem,” Math. Inequal. Appl., 11, No. 1, 75–80 (2007).
E. M. Semenov and B. S. Mitjagin, “Lack of interpolation of linear operators in spaces of smooth functions,” Mat. USSR. Izv., 11, 1229–1266 (1977).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 6, pp. 723–740, June, 2011.
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Acu, A.M., Gonska, H. & Raşa, I. Grüss-type and Ostrowski-type inequalities in approximation theory. Ukr Math J 63, 843–864 (2011). https://doi.org/10.1007/s11253-011-0548-2
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DOI: https://doi.org/10.1007/s11253-011-0548-2