| Title:
|
Bounds for Convex Functions of Čebyšev Functional Via Sonin's Identity with Applications (English) |
| Author:
|
Dragomir, Silvestru Sever |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
22 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
107-132 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Absolutely continuous functions |
| Keyword:
|
Convex functions |
| Keyword:
|
Integral inequalities |
| Keyword:
|
Čebyšev functional |
| Keyword:
|
Jensen's inequality |
| Keyword:
|
Lebesgue norms |
| Keyword:
|
Mid-point inequalities |
| Keyword:
|
Trapezoid inequalities |
| MSC:
|
25D10 |
| MSC:
|
26D15 |
| idZBL:
|
Zbl 1308.26030 |
| idMR:
|
MR3303133 |
| . |
| Date available:
|
2015-01-27T09:36:16Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144124 |
| . |
| Reference:
|
[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, Zbl 0996.26018, MR 1867810 |
| Reference:
|
[2] Cerone, P., Dragomir, S.S.: New bounds for the Čebyšev functional.App. Math. Lett., 18, 2005, 603-611, Zbl 1076.26017, MR 2131269, 10.1016/j.aml.2003.09.013 |
| Reference:
|
[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]. Zbl 1143.26009, MR 2321030 |
| Reference:
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[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, Zbl 1036.26018, MR 1982817, 10.1080/1065246031000074353 |
| Reference:
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[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, |
| Reference:
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[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]. Zbl 0994.26020, MR 1906398 |
| Reference:
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[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 |
| Reference:
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[8] Li, X., Mohapatra, R.N., Rodriguez, R.S.: Grüss-type inequalities.J. Math. Anal. Appl., 267, 2, 2002, 434-443, Zbl 1007.26016, MR 1888014, 10.1006/jmaa.2001.7565 |
| Reference:
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[9] Lupaş, A.: The best constant in an integral inequality.Mathematica (Cluj, Romania), 15 (38), 2, 1973, 219-222, Zbl 0285.26014, MR 0360960 |
| Reference:
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[10] Mercer, A.McD.: An improvement of the Grüss inequality.J. Inequal. Pure Appl. Math., 6, 4, 2005, Article 93, 4 pp. (electronic).. Zbl 1084.26014, MR 2178274 |
| Reference:
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[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 |
| Reference:
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[12] Ostrowski, A.M.: On an integral inequality.Aequat. Math., 4, 1970, 358-373, Zbl 0198.08106, MR 0268339, 10.1007/BF01844168 |
| Reference:
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