A generalization of multiple Wright-convex functions via randomization

In the paper we define and study classes W n ( Θ , M j ) of non-negative real functions associated with the classes M j 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 W n ( Θ , M j ) can be represented as a series of functions generated by a function from M j . 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, ⋂ n = 1 ∞ W n ( Θ , M j ) ⊋ M ∞ , and that when the Θ is exponential we have equality in the above formula.