A direct proof is given for an inequality relating the expected absolute value of stopped Brownian motion to the expected time to stopping. This inequality was originally proved by means of the martingale square function inequality. The latter is then derived from the former through use of a Skorokhod embedding. The first inequality is also applied to prove a martingale strong law of large numbers.
"An Equivalent to the Martingale Square Function Inequality." Ann. Math. Statist. 43 (6) 1927 - 1934, December, 1972. https://doi.org/10.1214/aoms/1177690863