Bounds for Convex Functions of Čebyšev Functional Via Sonin's Identity with Applications

Silvestru Sever Dragomir

Communications in Mathematics (2014)

  • Volume: 22, Issue: 2, page 107-132
  • ISSN: 1804-1388

Abstract

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Some new bounds for the Čebyšev functional in terms of the Lebesgue norms f - 1 b - a ∫ a b f ( t ) d t [ a , b ] , p and the Δ -seminorms ∥ f ∥ p Δ : = ∫ a b ∫ a b | f ( t ) - f ( s ) | p d t d s 1 p are established. Applications for mid-point and trapezoid inequalities are provided as well.

How to cite

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Dragomir, Silvestru Sever. "Bounds for Convex Functions of Čebyšev Functional Via Sonin's Identity with Applications." Communications in Mathematics 22.2 (2014): 107-132. <http://eudml.org/doc/269840>.

@article{Dragomir2014,
abstract = {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 \[ \Vert f\Vert \_\{p\}^\{\Delta \} := \biggl (\int \_a^b \int \_a^b |f(t)-f(s)|^\{p\}\{\,\mathrm \{d\}t\} \{\,\mathrm \{d\}s\} \biggr )^\{\frac\{1\}\{p\}\} \] are established. Applications for mid-point and trapezoid inequalities are provided as well.},
author = {Dragomir, Silvestru Sever},
journal = {Communications in Mathematics},
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; Chebyshev functional; Jensen's inequality; Lebesgue norms; mid-point inequalities; trapezoid inequalities},
language = {eng},
number = {2},
pages = {107-132},
publisher = {University of Ostrava},
title = {Bounds for Convex Functions of Čebyšev Functional Via Sonin's Identity with Applications},
url = {http://eudml.org/doc/269840},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Dragomir, Silvestru Sever
TI - Bounds for Convex Functions of Čebyšev Functional Via Sonin's Identity with Applications
JO - Communications in Mathematics
PY - 2014
PB - University of Ostrava
VL - 22
IS - 2
SP - 107
EP - 132
AB - 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 \[ \Vert f\Vert _{p}^{\Delta } := \biggl (\int _a^b \int _a^b |f(t)-f(s)|^{p}{\,\mathrm {d}t} {\,\mathrm {d}s} \biggr )^{\frac{1}{p}} \] are established. Applications for mid-point and trapezoid inequalities are provided as well.
LA - eng
KW - 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; Chebyshev functional; Jensen's inequality; Lebesgue norms; mid-point inequalities; trapezoid inequalities
UR - http://eudml.org/doc/269840
ER -

References

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  1. Cerone, P., Dragomir, S.S., Some bounds in terms of Δ -seminorms for Ostrowski-Grüss type inequalities, Soochow J. Math., 27, 4, 2001, 423-434, (2001) Zbl0996.26018MR1867810
  2. Cerone, P., Dragomir, S.S., 10.1016/j.aml.2003.09.013, App. Math. Lett., 18, 2005, 603-611, (2005) Zbl1076.26017MR2131269DOI10.1016/j.aml.2003.09.013
  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]. (2007) Zbl1143.26009MR2321030
  4. Cerone, P., Dragomir, S.S., Roumeliotis, J., 10.1080/1065246031000074353, Integral Transforms Spec. Funct., 14, 3, 2003, 205-216, (2003) Zbl1036.26018MR1982817DOI10.1080/1065246031000074353
  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, (1882) 
  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]. (2002) Zbl0994.26020MR1906398
  7. Grüss, G., Über das Maximum des absoluten Betrages von 1 b - a ∫ a b f ( x ) g ( x ) d x - 1 ( b - a ) 2 ∫ a b f ( x ) d x ∫ a b g ( x ) d x , Math. Z., 39, 1935, 215-226, (1935) MR1545499
  8. Li, X., Mohapatra, R.N., Rodriguez, R.S., 10.1006/jmaa.2001.7565, J. Math. Anal. Appl., 267, 2, 2002, 434-443, (2002) Zbl1007.26016MR1888014DOI10.1006/jmaa.2001.7565
  9. Lupaş, A., The best constant in an integral inequality, Mathematica (Cluj, Romania), 15 (38), 2, 1973, 219-222, (1973) Zbl0285.26014MR0360960
  10. Mercer, A.McD., An improvement of the Grüss inequality, J. Inequal. Pure Appl. Math., 6, 4, 2005, Article 93, 4 pp. (electronic).. (2005) Zbl1084.26014MR2178274
  11. Mitrinović, D.S., Pečarić, J.E., Fink, A.M., Classical and New Inequalities in Analysis, 1993, Kluwer Academic Publishers, Dordrecht/Boston/London, (1993) MR1220224
  12. Ostrowski, A.M., 10.1007/BF01844168, Aequat. Math., 4, 1970, 358-373, (1970) Zbl0198.08106MR0268339DOI10.1007/BF01844168
  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.. (2005) Zbl1088.26017MR2164323

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