On some estimates for bounded submartingales and the shift inequality

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