Naslov
Superadditivity of the Jensen-type functionals and applications

Autori
Krnić, Mario

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Mathematical Inequalities and Applications, MIA 2014 / - , 2014

Skup
One thousand papers conference, MIA 2014, Trogir

Mjesto i datum
Trogir, Hrvatska, 22.06.2014. - 26.06.2014

Vrsta sudjelovanja
Plenarno

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Jensen inequality; Jensen functional; superadditivity

Sažetak
By a discrete Jensen functional we mean the difference between the right-hand side and the left-hand side of the inequality $$ P_{;n};f\left(\frac{;1};{;P_{;n};};\sum_{;i=1};^{;n};p_{;i};x_{;i};\right)\leq \sum_{;i=1};^{;n};p_{;i};f(x_{;i};), $$ where $f:I\rightarrow \mathbb{;R};$ is a convex function, $p_i>0$, $x_i\in I$, $i=1, 2, \ldots , n$, and $P_n=\sum_{;i=1};^n p_i$. It has been shown in 1996 that the discrete Jensen functional possesses the properties of superadditivity and monotonicity on the set of non-negative real $n$-tuples (S.S. Dragomir, J.E. Pe\v{;c};ari\'{;c}; and L.E. Persson). In a similar sense, functionals associated with the inequalities related to the Jensen inequality are observed in this talk. They are proved to possess the analogous properties of superadditivity and monotonicity on suitable sets of the weighted real valued functions (the Jessen and the McShane functional) or real $n$-tuples (the Jensen-Steffensen and the Jensen-Mercer functional). Consequently, their lower and upper bounds expressed by non-weighted functionals of the same type are obtained. In such a way, refinements and converses of the starting inequalities are obtained. Their further applications are manifested in the refined and conversed forms of numerous other inequalities, e.g. inequalities for the generalized weighted\- and power means, the Young inequality, the H\"{;o};lder inequality, the Hilbert inequality and so on. In the second part of the talk, real arguments of the Jensen-type functionals are substituted by the bounded selfadjoint operators on a Hilbert space. The general method for refinements and converses of the inequalities for certain operator means (arithmetic, geometric and Heinz means) is developed, due to the properties of superadditivity and monotonicity, which corresponding functionals are proved to possess as well. Furthermore, integral analogues of the above mentioned properties are obtained by means of the Jensen operator inequality. With the notion of convexity expanded to the operator functions of several variables, described considerations are carried out for the multidimensional Jensen functional. Consequently, refinements and converses of the weighted operator inequalities of H\"{;o};lder and Minkowski are obtained. By means of the Jensen functional, refinements of Heinz norm inequalities are also derived. Finally, some eigenvalue and weak majorization inequalities related to the Jensen inequality are presented.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA