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Two Inequalities for the First Moments of a Martingale, its Square Function and its Maximal Function
Given a Hilbert space valued martingale $(M_n)$, let $(M^*_n)$ and $(S_n(M))$ denote its maximal function and square function, respectively. We prove that $$\displaylines{ \mathbb{E}|M_n|\leq 2\mathbb{E}S_n(M), \quad\ n=0,1,2,\ldots,\cr \mathbb{E}M^*_