# zbMATH — the first resource for mathematics

Moyennes limites et convexité. (Limit mean values and convexity). (French) Zbl 0673.26004
Let F be the set of bounded functions defined on $${\mathbb{R}}_+$$, taking nonnegative real values, and Lebesgue-integrable on every interval [0,t]. To each function a in F we associate the ordered pair L(a) of the upper and lower limits of $$t^{-1}\int^{t}_{0}a(s)ds$$, as t tends to $$+\infty$$. We characterize the set $S(a)=\{L(b)\in {\mathbb{R}}^ 2;\quad b\in F\quad and\quad \forall t\in {\mathbb{R}}_+,\quad b(t)\leq a(t)\}$ as being a closed convex region of $${\mathbb{R}}^ 2$$.
Reviewer: G.Grekos

##### MSC:
 26A42 Integrals of Riemann, Stieltjes and Lebesgue type 52A10 Convex sets in $$2$$ dimensions (including convex curves) 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems 28A25 Integration with respect to measures and other set functions
Full Text:
##### References:
 [1] G.Grekos, Répartition des densités des sous-suites d’une suite d’entiers,J. Number Theory,10 (1978), 177–191.MR 58: 22006;Zbl. 388: 10033 · Zbl 0388.10033 · doi:10.1016/0022-314X(78)90034-3 [2] G.Grekos and B.Volkmann, On densities and gaps,J. Number Theory,26 (1987), 129–148.Zbl. 622: 10044.MR 88: 11009 · Zbl 0622.10044 · doi:10.1016/0022-314X(87)90074-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.