A new proof of a sharp $L^p$-inequality for stochastic integrals is given that makes it possible to show that strict inequality holds in all nontrivial cases.
"A Sharp and Strict $L^p$-Inequality for Stochastic Integrals." Ann. Probab. 15 (1) 268 - 273, January, 1987. https://doi.org/10.1214/aop/1176992268