A convexity-type functional inequality with infinite convex combinations

Given a function f defined on a nonempty and convex subset of the d-dimensional Euclidean space, we prove that if f is bounded from below and it satisfies a convexity-type functional inequality with infinite convex combinations, then f has to be convex. We also give alternative proofs of a generalization of some known results on convexity with infinite convex combinations due to Dar´oczy and P´ales (1987) and Pavi´c (2019) using a probabilistic version of Jensen inequality.