Naslov
Jensen-Steffensen inequality: old and new

Autori
Klaričić Bakula, Milica

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

Izvornik
Conference on Inequalities and Applications 2016 / Gilányi, Attila ; Boros, Zoltán, Bessenyei, Mihály - Deberecen : University of Debrecen, 2016, 33-33

Skup
Conference on Inequalities and Applications 2016

Mjesto i datum
Hajdúszoboszló, Mađarska, 28.08.2016. - 03.09.2016

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Jensen-Steffensen inequality, generalized convexity

Sažetak
{;Let $I$ be an interval in $\mathbb{;R};$ and $f:I\rightarrow \mathbb{;R};$ a convex function on $I$.\ If $\boldsymbol{;\xi };=\left( \xi _{;1};, \cdots , \xi _{;m};\right) $ is any $m$-tuple in $I^{;m};$ and $\boldsymbol{;p};=\left( p_{;1};, \cdots , p_{;m};\right) $ any nonnegative $m$-tuple such that $% \sum_{;i=1};^{;m};p_{;i};>0$, then the well known Jensen's inequality \begin{;equation}; f\left( \frac{;1};{;P_{;m};};\sum_{;i=1};^{;m};p_{;i};\xi _{;i};\right) \leq \frac{;1};{;P_{;m};% };\sum_{;i=1};^{;m};p_{;i};f\left( \xi _{;i};\right) \label{;jen}; \end{;equation};% holds, where $P_{;m};=\sum_{;i=1};^{;m};p_{;i};$ . It is known that the assumption \textquotedblright $\boldsymbol{;p};$ is a nonnegative $m$-tuple\textquotedblright\ can be relaxed at the expense of more restrictions on the $m$-tuple $\boldsymbol{;\xi };$. Namely, if $% \boldsymbol{;p};$ is a real $m$-tuple such that \begin{;equation}; 0\leq P_{;j};\leq P_{;m};\text{; };, \text{; };j=1, \cdots , m\text{; }; ; \ \ \ \ P_{;m};>0% \text{; };, \label{;je-st}; \end{;equation};% where $P_{;j};:=\sum_{;i=1};^{;j};p_{;i};$ , then for any monotonic $m$-tuple $% \boldsymbol{;\xi };$\ (increasing or decreasing) in $I^{;m};$ we get \[ \overline{;\xi };=\frac{;1}; {;P_{;m};};\sum_{;i=1};^{;m};p_{;i};\xi _{;i};\in I\text{; };, \]% and for any function $f$ convex on $I, $ $\left( \ref{;jen};\right) $ still holds. Inequality $\left( \ref{;jen};\right) $ considered under conditions $% \left( \ref{;je-st};\right) $ is known as the Jensen-Steffensen inequality for convex functions. We can say that the Jensen- Steffensen inequality is "the ugly sister" of Jensen's inequality: not much admired and usually "not invited to the party". Our goal here is to show that "she" has many hidden beauties and that "she" can proudly walk hand in hand with her well known sister.};

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA