Sharp moment estimates for martingales with uniformly bounded square functions

We provide sharp bounds for the exponential moments and $p$-moments, $1\leqslant p \leqslant 2$, of terminate distribution a martingale whose square function is uniformly bounded by one. introduce Bellman corresponding extremal problem reduce it to already known on $\mathrm{BMO}([0,1])$. In case tail estimates, similar reduction does not work exactly, so we come up with fine supersolution that leads estimates.