Abstract
Let $X$ be an $L_p$ martingale, $3 \leqslant p < \infty$. Let $M = \sup|X_k|$ and $V^2 = \Sigma(X_k - X_{k-1})^2$. We show that $\|X\|_p \leqslant (p - 1)\|V\|_p$ and, consequently, that $\|M\|_p \leqslant p\|V\|_p$.
Citation
A. O. Pittenger. "Note on a Square Function Inequality." Ann. Probab. 7 (5) 907 - 908, October, 1979. https://doi.org/10.1214/aop/1176994952
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