Sharp bounds for distribution of martingale transform of bounded functions

The paper suggests a characterization of positive function $ f $ such that $ f(\psi_\infty) $ is summable for any limit value $ \psi_\infty $ of the martingale transform for an indicator function. The characterizing condition is that a version of the Lipschitz majorant of $ f $ is summable with respect to an exponential weight. The reasoning is based on the calculation of particular minimal biconcave functions on the strip.