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Keywords:
Absolutely continuous functions; Convex functions; Integral inequalities; Čebyšev functional; Jensen's inequality; Lebesgue norms; Mid-point inequalities; Trapezoid inequalities
Absolutely continuous functions; Convex functions; Integral inequalities; Čebyšev functional; Jensen's inequality; Lebesgue norms; Mid-point inequalities; Trapezoid inequalities
Summary:
Some new bounds for the Čebyšev functional in terms of the Lebesgue norms $$ \biggl \Vert f-\frac {1}{b-a}\int _a^b f(t){\,\mathrm{d}t} \biggr \Vert _{[a,b],p} $$ and the $\Delta $-seminorms $$ \lVert f\rVert _{p}^{\Delta } := \biggl (\int _a^b \int _a^b |f(t)-f(s)|^{p}{\,\mathrm{d}t} {\,\mathrm{d}s} \biggr )^{\frac 1p} $$ are established. Applications for mid-point and trapezoid inequalities are provided as well.
References:
[1] Cerone, P., Dragomir, S.S.: Some bounds in terms of $\Delta $-seminorms for Ostrowski-Grüss type inequalities. Soochow J. Math., 27, 4, 2001, 423-434, MR 1867810 | Zbl 0996.26018
[2] Cerone, P., Dragomir, S.S.: New bounds for the Čebyšev functional. App. Math. Lett., 18, 2005, 603-611, DOI 10.1016/j.aml.2003.09.013 | MR 2131269 | Zbl 1076.26017
[3] Cerone, P., Dragomir, S.S.: A refinement of the Grüss inequality and applications. Tamkang J. Math., 38, 1, 2007, 37-49, Preprint RGMIA Res. Rep. Coll. 5 (2) (2002), Art. 14. [online http://rgmia.vu.edu.au/v8n2.html] MR 2321030 | Zbl 1143.26009
[4] Cerone, P., Dragomir, S.S., Roumeliotis, J.: Grüss inequality in terms of $\Delta $-seminorms and applications. Integral Transforms Spec. Funct., 14, 3, 2003, 205-216, DOI 10.1080/1065246031000074353 | MR 1982817 | Zbl 1036.26018
[5] Chebyshev, P.L.: Sur les expressions approximatives des intègrals dèfinis par les outres prises entre les même limites. Proc. Math. Soc. Charkov, 2, 1882, 93-98,
[6] Cheng, X.-L., Sun, J.: Note on the perturbed trapezoid inequality. J. Ineq. Pure & Appl. Math., 3, 2, 2002, Art. 29. [online http://jipam.vu.edu.au/article.php?sid=181] MR 1906398 | Zbl 0994.26020
[7] Grüss, G.: Über das Maximum des absoluten Betrages von $\frac{1}{b-a}\int_{a}^{b}f(x)g(x)\,{\rm d}x-\frac{1}{(b-a)^{2}}\int_{a}^{b}f(x)\,{\rm d}x\int_{a}^{b}g(x)\,{\rm d}x$. Math. Z., 39, 1935, 215-226, MR 1545499
[8] Li, X., Mohapatra, R.N., Rodriguez, R.S.: Grüss-type inequalities. J. Math. Anal. Appl., 267, 2, 2002, 434-443, DOI 10.1006/jmaa.2001.7565 | MR 1888014 | Zbl 1007.26016
[9] Lupaş, A.: The best constant in an integral inequality. Mathematica (Cluj, Romania), 15 (38), 2, 1973, 219-222, MR 0360960 | Zbl 0285.26014
[10] Mercer, A.McD.: An improvement of the Grüss inequality. J. Inequal. Pure Appl. Math., 6, 4, 2005, Article 93, 4 pp. (electronic).. MR 2178274 | Zbl 1084.26014
[11] Mitrinović, D.S., Pečarić, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. 1993, Kluwer Academic Publishers, Dordrecht/Boston/London, MR 1220224
[12] Ostrowski, A.M.: On an integral inequality. Aequat. Math., 4, 1970, 358-373, DOI 10.1007/BF01844168 | MR 0268339 | Zbl 0198.08106
[13] Pachpatte, B.G.: On Grüss like integral inequalities via Pompeiu's mean value theorem. J. Inequal. Pure Appl. Math., 6, 3, 2005, Article 82, 5 pp.. MR 2164323 | Zbl 1088.26017
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