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Extensions and improvements of Sherman’s and related inequalities forn-convex functions

100%
Bradanović S. I., Pečarić J.
Open Mathematics
|
2017
|
tom 15
|
nr 1
936-947
EN
This paper gives extensions and improvements of Sherman’s inequality for n-convex functions obtained by using new identities which involve Green’s functions and Fink’s identity. Moreover, extensions and improvements of Majorization inequality as well as Jensen’s inequality are obtained as direct consequences. New inequalities between geometric, logarithmic and arithmetic means are also established.
2
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Improved Heinz inequalities via the Jensen functional

100%
Krnić M., Pečarić J.
Open Mathematics
|
2013
|
tom 11
|
nr 9
1698-1710
EN
By virtue of convexity of Heinz means, in this paper we derive several refinements of Heinz norm inequalities with the help of the Jensen functional and its properties. In addition, we discuss another approach to Heinz operator means which is more convenient for obtaining the corresponding operator inequalities for positive invertible operators.
3
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Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions

81%
Abramovich S., Ivelić S., Pečarić J.
Open Mathematics
|
2010
|
tom 8
|
nr 5
937-949
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.
4
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Generalizations of the Jensen-Steffensen and related inequalities

81%
Bakula M., Matić M., Pečarić J.
Open Mathematics
|
2009
|
tom 7
|
nr 4
787-803
EN
We present a couple of general inequalities related to the Jensen-Steffensen inequality in its discrete and integral form. The Jensen-Steffensen inequality, Slater’s inequality and a generalization of the counterpart to the Jensen-Steffensen inequality are deduced as special cases from these general inequalities.
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