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• # Artykuł - szczegóły

## Open Mathematics

2010 | 8 | 5 | 937-949

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

EN

### Abstrakty

EN
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.

EN

937-949

wydano
2010-10-01
online
2010-09-28

### Twórcy

autor
• University of Haifa
autor
• University of Split
autor
• University of Zagreb

### Bibliografia

• [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
• [2] Abramovich S., Jameson G., Sinnamon G., Refining Jensen’s inequality, Bull. Math. Soc. Sci. Math. Roumanie, 2004, 47(95)(1–2), 3–14
• [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
• [4] Boas R.P., The Jensen-Steffensen inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., 1970, 302–319, 1–8
• [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
• [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
• [7] Mercer A.McD., A variant of Jensen’s inequality, JIPAM. J. Inequal. Pure Appl. Math., 2003, 4(4), article 73
• [8] Pečarić J.E., Proschan F., Tong Y.L., Convex Functions, Partial Orderings, and Statistical Applications, Math. Sci. Eng., 187, Academic Press, Boston, 1992