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

Shoshana Abramovich; Slavica Ivelić; Josip Pečarić

Open Mathematics (2010)

- Volume: 8, Issue: 5, page 937-949
- ISSN: 2391-5455

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topShoshana Abramovich, Slavica Ivelić, and Josip Pečarić. "Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions." Open Mathematics 8.5 (2010): 937-949. <http://eudml.org/doc/269389>.

@article{ShoshanaAbramovich2010,

abstract = {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 \[ \bar\{y\} \geqslant \phi \left( \{\bar\{x\}\} \right) + \frac\{1\}\{\{\lambda \left( \beta \right) - \lambda \left( \alpha \right)\}\}\int \_\alpha ^\beta \{\phi \left( \{\left| \{f\left( t \right) - \bar\{x\}\} \right|\} \right)d\lambda \left( t \right)\} \]
, where \[ \bar\{x\} = \frac\{1\}\{\{\lambda \left( \beta \right) - \lambda \left( \alpha \right)\}\}\int \_\alpha ^\beta \{f\left( t \right)d\lambda \left( t \right)\} \]
and \[ \bar\{y\} = \frac\{1\}\{\{\lambda \left( \beta \right) - \lambda \left( \alpha \right)\}\}\int \_\alpha ^\beta \{\phi \left( \{f\left( t \right)\} \right)d\lambda \left( t \right)\} \]
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.},

author = {Shoshana Abramovich, Slavica Ivelić, Josip Pečarić},

journal = {Open Mathematics},

keywords = {Jensen-Steffensen inequality; Convex functions; Superquadratic functions; convex functions; superquadratic functions},

language = {eng},

number = {5},

pages = {937-949},

title = {Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions},

url = {http://eudml.org/doc/269389},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Shoshana Abramovich

AU - Slavica Ivelić

AU - Josip Pečarić

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

JO - Open Mathematics

PY - 2010

VL - 8

IS - 5

SP - 937

EP - 949

AB - 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 \[ \bar{y} \geqslant \phi \left( {\bar{x}} \right) + \frac{1}{{\lambda \left( \beta \right) - \lambda \left( \alpha \right)}}\int _\alpha ^\beta {\phi \left( {\left| {f\left( t \right) - \bar{x}} \right|} \right)d\lambda \left( t \right)} \]
, where \[ \bar{x} = \frac{1}{{\lambda \left( \beta \right) - \lambda \left( \alpha \right)}}\int _\alpha ^\beta {f\left( t \right)d\lambda \left( t \right)} \]
and \[ \bar{y} = \frac{1}{{\lambda \left( \beta \right) - \lambda \left( \alpha \right)}}\int _\alpha ^\beta {\phi \left( {f\left( t \right)} \right)d\lambda \left( t \right)} \]
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.

LA - eng

KW - Jensen-Steffensen inequality; Convex functions; Superquadratic functions; convex functions; superquadratic functions

UR - http://eudml.org/doc/269389

ER -

## References

top- [1] Abramovich S., Ivelić S., Pečarić J., Improvement of Jensen-Steffensen’s inequality for superquadratic functions, Banach J. Math. Anal., 2010, 4(1), 159–169 Zbl1195.26037
- [2] Abramovich S., Jameson G., Sinnamon G., Refining Jensen’s inequality, Bull. Math. Soc. Sci. Math. Roumanie, 2004, 47(95)(1–2), 3–14 Zbl1150.26333
- [3] Abramovich S., Jameson G., Sinnamon G., Inequalities for averages of convex and superquadratic functions, JIPAM. J. Inequal. Pure Appl. Math., 2004, 5(4), article 91 Zbl1057.26009
- [4] Boas R.P., The Jensen-Steffensen inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 1970, 302–319, 1–8 Zbl0213.34601
- [5] Klaričić Bakula M., Matić M., Pečarić J., On some general inequalities related to Jensen’s inequality, In: International Series of Numerical Mathematics, 157, Inequalities and Applications, Conference on Inequalities and Applications, Noszvaj (Hungary), September 2007, Birkhäuser, Basel, 2008, 233–243 Zbl1266.26044
- [6] Klaričić Bakula M., Matić M., Pečarić J., Generalizations of the Jensen-Steffensen and related inequalities, Cent. Eur. J. Math., 2009, 7(4), 787–803 http://dx.doi.org/10.2478/s11533-009-0052-1 Zbl1183.26020
- [7] Mercer A.McD., A variant of Jensen’s inequality, JIPAM. J. Inequal. Pure Appl. Math., 2003, 4(4), article 73 Zbl1048.26016
- [8] Pečarić J.E., Proschan F., Tong Y.L., Convex Functions, Partial Orderings, and Statistical Applications, Math. Sci. Eng., 187, Academic Press, Boston, 1992 Zbl0749.26004

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