engleski

Jensen-Steffensen inequality: old and new

{;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.};

Jensen-Steffensen inequality, generalized convexity

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