On a generalization of the class of Jensen convex functions

Abstract

The main objective of this article is to introduce a new class of real valued functions that include the well-known class of \({m}\)-convex functions introduced by Toader (1984). The members of this collection are called Jensen \({m}\)-convex and are defined, for \({m \in (0,1]}\), via the functional inequality $$f\left(\frac{x+y}{c_m} \right)\leq \frac{f(x)+f(y)}{c_m} \quad(x,y \in [0,b]),$$ where \({c_{m} := \frac{m+1}{m}}\). These functions generate a new kind of functional convexity that is studied in terms of its behavior with respect to basic algebraic operations such as sums, products, compositions, etc. in this paper. In particular, it is proved that any starshaped Jensen convex function is Jensen \({m}\)-convex. At the same time an interesting example (Example 3) shows how the classes of Jensen \({m}\)-convex functions depend on \({m}\). All the techniques employed come from traditional basic calculus and most of the calculations have been done with Mathematica 8.0.0 and validated with Maple 15 as well as all the figures included.