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Sharp inequality for bounded submartingales and their differential subordinates. (English) Zbl 1196.60074
Summary: Let $$\alpha$$ be a fixed number from the interval $$[0,1]$$. We obtain the sharp probability bounds for the maximal function of the process which is $$\alpha$$-differentially subordinate to a bounded submartingale. This generalizes the previous results of D. L. Burkholder [Calcul des probabilités, Ec. d’Été, Saint-Flour/Fr. 1989, Lect. Notes Math. 1464, 1–66 (1991; Zbl 0771.60033)] and W. Hammack [Ann. Probab. 23, No. 1, 223–235 (1995; Zbl 0833.60057)].

##### MSC:
 60G42 Martingales with discrete parameter 60G46 Martingales and classical analysis
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