Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions

Shoshana Abramovich; Slavica Ivelić; Josip Pečarić

Shoshana Abramovich ; Slavica Ivelić ; Josip Pečarić

Open Mathematics, Tome 8 (2010), p. 937-949 / Harvested from The Polish Digital Mathematics Library

Résumé

We present integral versions of some recently proved results which improve the Jensen-Steffensen and related inequalities for superquadratic functions. For superquadratic functions which are not convex we get inequalities analogous to the integral Jensen-Steffensen inequality for convex functions. Therefore, we get refinements of all the results which use only the convexity of these functions. One of the inequalities that we obtain for a superquadratic function φ is $\bar{y} ⩾ φ (\bar{x}) + \frac{1}{λ (β) - λ (α)} \int_{α}^{β} φ (|f (t) - \bar{x}|) d λ (t)$ , where $\bar{x} = \frac{1}{λ (β) - λ (α)} \int_{α}^{β} f (t) d λ (t)$ and $\bar{y} = \frac{1}{λ (β) - λ (α)} \int_{α}^{β} φ (f (t)) d λ (t)$ which under suitable conditions like those satisfied by functions of power equal or more than 2, is a refinement of the Jensen-Steffensen-Boas inequality. We also prove related results of Mercer’s type.

Publié le : 2010-01-01

Zbl 1217.26038

EUDML-ID : urn:eudml:doc:269389

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     author = {Shoshana Abramovich and Slavica Iveli\'c and Josip Pe\v cari\'c},
     title = {Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions},
     journal = {Open Mathematics},
     volume = {8},
     year = {2010},
     pages = {937-949},
     zbl = {1217.26038},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0055-y}
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Shoshana Abramovich; Slavica Ivelić; Josip Pečarić. Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions. Open Mathematics, Tome 8 (2010) pp. 937-949. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0055-y/

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