We unify several Bellman function problems treated in [1,2,4-6,9-12,14-16,18-25]. For that purpose, we define a class of functions that have, in a sense, small mean oscillation (this class depends on two convex sets in R-2). We show how the unit ball in the BMO space, or a Muckenhoupt class, or a Gehring class can be described in such a fashion. Finally, we consider a Bellman function problem on these classes, discuss its solution and related questions. (C) 2015 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.