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

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Abramovich S., Ivelić S., Pečarić J.

Open Mathematics

2010

tom 8

nr 5

937-949

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.

Ograniczanie wyników

1 Open Mathematics

1 Abramovich S.

1 Ivelić S.

1 Pečarić J.

1 2010

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