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It is well known that if a submartingale $X$ is bounded then the increasingpredictable process $Y$ and the martingale $M$ from the Doob decomposition $%X=Y+M$ can be unbounded. In this paper for some classes of increasing convexfunctions $f$ we will find the upper bounds for $\lim_n\sup_XEf(Y_n)$, wherethe supremum is taken over all submartingales $(X_n),0\leq X_n\leq1,n=0,1,...$. We apply the stochastic control theory to prove these results.

On some estimates for bounded submartingales and the shift inequality