  We consider the strong form of the John-Nirenberg inequality for the
$L^2$-based BMO. We construct explicit Bellman functions for the inequality in
the continuous and dyadic settings and obtain the sharp constant as well as the
precise bound on the inequality's range of validity, both previously unknown.
The results for the two cases are substantially different. The paper not only
gives another instance in the short list of such explicit calculations, but
also presents the Bellman function method as a sequence of clear steps,
adaptable to a wide variety of applications.
