Rearrangements, L-Superadditivity and Jensen-Type Inequalities

Abstract

We deal here with the minimum and the maximum of

$$\displaystyle\sum _{i=1}^{n}F\left ( a_{2i-1},a_{2i}\right ) ,\left ( \mathbf {a}\right ) \in \mathbb {R} ^{2n}$$

and of

$$\displaystyle\sum _{i=1}^{n}F\left ( a_{i},a_{i+1}\right ) ,\ \ a_{n+1}=a_{1},\left ( \mathbf {a}\right ) \in \mathbb {R} ^{n}$$

obtained by using rearrangement techniques. The results depend on the arrangement of $\left ( \mathbf {a}\right )$ and are used in proving Jensen-type inequalities.