A class of means and related inequalities
The author defines mean values \(M_ u\) as the unique solutions of \[ \frac{u(M_ u(a_ 1,...,a_ n))}{M_ u(a_ 1,...,a_ n)}=\sum^{n}_{i=1}u(\prod_{j\neq i}a_ j)/\sum^{n}_{i=1}a_ i \] where \(u: I\to {\mathbb{R}}_+\) is continuous and either decreasing or such that \(x\mapsto u(x)/x\) is strictly increasing; I is a real interval such that \(x\in I\) implies \(x^ n\in I\) for all positive integers n. The arithmetic, the geometric and several other means fall into this category, but the harmonic mean does not [cf. the reviewer, C. R. Math. Rep. Acad. Sci., R. Can. 6, 319-336 (1984; Zbl 0563.39003); the concept originates in Korovkin approximation]. The author shows that, if u is also convex, then \(M_ u\) is not larger than the arithmetic mean. The equality cases are also determined.
strictly positive continuous real-valued functions
scientific article; zbMATH DE number 3950902