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We unify several Bellman function problems into one setting. For that purposewe 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 unitball in the $\mathrm{BMO}$ space, or a Muckenhoupt class, or a Gehring classcan be described in such a fashion. Finally, we consider a Bellman functionproblem on these classes, discuss its solution and related questions.

Sharp estimates of integral functionals on classes of functions with small mean oscillation