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 Daróczy and Páles (1987) and Pavić (2019) using a probabilistic version of Jensen inequality.