Abstract
If $g$ is the transform of a martingale $f$ under a predictable sequence $v$ uniformly bounded in absolute value by 1, then $$\lambda P(g^\ast \geqslant \lambda) \leqslant 2\|f\|_1, \lambda > 0$$, and this inequality is sharp.
Citation
D. L. Burkholder. "A Sharp Inequality for Martingale Transforms." Ann. Probab. 7 (5) 858 - 863, October, 1979. https://doi.org/10.1214/aop/1176994944
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