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 Burkholder and Hammack.
"Sharp inequality for bounded submartingales and their differential subordinates." Electron. Commun. Probab. 13 660 - 675, 2008. https://doi.org/10.1214/ECP.v13-1433