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

Shoshana 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 -