An inequality for certain random sequences more general than martingales or nonnegative submartingales is proved. Three special cases are deduced, one of which generalizes and refines a result of Austin. As an application of the inequality, the special cases are used to give new proofs of Burkholder's $L \log L$ and $L_p$ (for $1 < p \leqslant 2$) inequalities for the square function of a martingale or a nonnegative submartingale.
"A Martingale Inequality for the Square and Maximal Functions." Ann. Probab. 7 (6) 1051 - 1055, December, 1979. https://doi.org/10.1214/aop/1176994898