We give a review of results proved and published mostly in recent years, concerning real-valued convex functions as well as almost convex functions defined on a (not necessarily convex) subset of a group. Analogues of such classical results as the theorems of Jensen, Bernstein-Doetsch, Blumberg-Sierpiński, Ostrowski, and Mehdi are presented. A version of the Hahn-Banach theorem with a convex control function is proved, too. We also study some questions specific for the group setting, for instance...

We investigate the criticality of the one term $2n$-order difference operators $l{\left(y\right)}_k={\Delta }^n\left(r_k{\Delta }^n{y}_k\right)$. We explicitly determine the recessive and the dominant system of solutions of the equation $l{\left(y\right)}_k=0$. Using their structure we prove a criticality criterion.