Abstract
In this paper we study Grüss type inequalities for real and complex valued functions in probability spaces. Some earlier Grüss type inequalities are extended and refined. Our approach leads to new integral inequalities which are interesting in their own right. As an application, we give a Grüss type inequality for normal operators in a Hilbert space. Similar results are obtained only for self-adjoint operators in earlier papers.
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Cerone, P., Dragomir, S.S.: A refinement of the Grüss inequality and applications. Tamkang J. Math. 38, 37–49 (2007)
Cheng, X. L., Sun, J.: A note on the perturbed trapezoid inequality. J. Ineq. Pure. Appl. Math. 3 (2002) Article 29
Dragomir, S.S.: A generalization of Grüss’s inequality in inner product spaces and applications. J. Math. Anal. Appl. 237, 74–82 (1999)
Grüss, G.: Über das Maximum des absoluten Betrages von \(\frac{1}{b-a}\int _{a}^{b}f\left( x\right) g\left( x\right) dx-\frac{1}{b-a}\int _{a}^{b}f\left( x\right) dx\frac{1}{b-a}\int _{a}^{b}g\left( x\right) dx\). Math. Z. 39, 215–226 (1935)
Li, Xin, Mohapatra, R.N., Rodriguez, R.S.: Grüss-type inequalities. J. Math. Anal. Appl. 267, 434–443 (2002)
Mitrinović, D., Pečarić, J., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht (1993)
Mond, B., Pečarić, J.: On some operator inequalities. Indian J. Math. 35, 221–232 (1993)
Rudin, W.: Functional Analysis. McGraw-Hill, Inc, New York (1991)
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The research of the author has been supported by Hungarian National Foundations for Scientific Research Grant No. K120186.
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Horváth, L. Grüss type and related integral inequalities in probability spaces. Aequat. Math. 93, 743–756 (2019). https://doi.org/10.1007/s00010-018-0612-1
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DOI: https://doi.org/10.1007/s00010-018-0612-1