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With this paper, we begin a series of studies of extremal problems forestimating distributions of martingale transforms of bounded martingales. TheBellman functions corresponding to such problems are pointwise minimaldiagonally concave functions on a horizontal strip, satisfying certain givenboundary conditions. We describe the basic structures that arise whenconstructing such functions and present a solution in the case of asymmetricboundary conditions and a sufficiently small width of the strip.

Some extremal problems for martingale transforms, I