On probability and moment inequalities for supermartingales and martingales

The probability inequality for $\max_{k\le n}S_k$, where $S_k=\sum_{j=1}^kX_j$, is proved under the assumption that the sequence $S_k$, $k=1,\dots,n$ is a supermartingale. This inequality is stated in terms of probabilities $\mathbf P(X_j>y)$ and conditional variances of random variables $X_j$, $j=1,\dots,n$. As a simple consequence the well-known moment inequality due to Burkholder is deduced. Numerical bounds are given for constants in Burkholder's inequality.