A generalization of multiple Wright-convex functions via randomization

In the paper we define and study classes Wn(Θ,Mj) of non-negative real functions associated with the classes Mj of j-times monotone functions. These classes are generalizations of n-Wright-convex functions introduced in Gilányi and Páles (2008) [2], and studied by Maksa and Páles (2009) [6]. We prove that each function from Wn(Θ,Mj) can be represented as a series of functions generated by a function from Mj. We give an integral representation of these functions in the case when a random variable Θ has an exponential or a discrete arithmetic distribution. As a consequence we show, that for an arithmetic discrete Θ, , and that when the Θ is exponential we have equality in the above formula.