  We unify several Bellman function problems into one setting. 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 $\mathbb{R}^2$). We show how the unit
ball in the $\mathrm{BMO}$ 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.
