Three stories about a Bellman function

I will survey some sharp inequalities for BMO functions obtained using a special form of optimization called the Bellman funciton method. After that I will show how the already computed Bellman functions provide solutions to two unrelated optimization problems in probability theory. One of these problems concerns sharp estimates for the distribution of the martinagle transform of an event, the other asks for sharp estimates for the distribution of a martingale whose square function is bounded.