Bellman function for a parametric family of extremal problems in $\mathrm{BMO}$

Suppose $I$ is an interval on the real line and $\langle\cdot\rangle_I$ is the corresponding integral average. We describe how the Bellman function for the functional $F(\varphi)=\langle f\circ\varphi\rangle_I$, $\varphi\in\mathrm{BMO}(I)$, varies when $f$ runs over a certain parametric family of functions. Thereby, we once again demonstrate the work of the methods developed recently by V. I. Vasyunin, P. B. Zatitskiy, P. Ivanishvili, D. M. Stolyarov, and the author.