Let $f=(f_n)$ be a nonnegative submartingale such that ${parallel}f{parallel}{infty}{leq}1;and;g=(g_n)$ be a martingale, adapted to the same filtration, satisfying $${mid}d_{gn}{mid}{leq}{mid}df_n{mid},;n=0,;1,;2,;....$$ The paper contains the proof of the sharp inequality $$limits^{sup}_ n;mathbb{E}{Phi}({mid}g_n{mid}){leq}{Phi}(1)$$ for a class of convex increasing functions ${Phi};on;[0,;{infty}]$, satisfying certain growth condition. As an application, we show a continuous-time version for stochastic integrals and a related estimate for smooth functions on Euclidean domain.