  With this paper, we begin a series of studies of extremal problems for
estimating distributions of martingale transforms of bounded martingales. The
Bellman functions corresponding to such problems are pointwise minimal
diagonally concave functions on a horizontal strip, satisfying certain given
boundary conditions. We describe the basic structures that arise when
constructing such functions and present a solution in the case of asymmetric
boundary conditions and a sufficiently small width of the strip.
