The best constant in the Davis inequality for the expectation of the martingale square function (Q2750945)

Let \((\Omega,{\mathcal F},\mathbb{F}, P)\) be a stochastic basis, \(\{f_n:= \sum^n_{k=0} d_k\), \(n\geq 0\}\) be a Hilbert space \((H,|\cdot|)\)-valued \((P,\mathbb{F})\)-martingale, \(f^*= \sup_{n\geq 0}|f_n|\), \(S(f)= \lim_{n\to\infty} [\sum^n_{k=0}|d_k|^2]^{1/2}\). It is proved that if \(1\leq p\leq 2\), then NEWLINE\[NEWLINE\|S(f)\|_p\leq \gamma_p\|f^*\|_p,NEWLINE\]NEWLINE where \(\|\cdot\|\) is an \(L_p\) norm and NEWLINE\[NEWLINE\gamma^2_p= 1+ \sup_{y> 1} {(2y- 1)(1- y^{p-2})\over y^p-1}.NEWLINE\]NEWLINE If \(p=1\), then the constant \(\gamma_1= \sqrt 3\) is the best possible. An analogous result is proved for continuous time \(H\)-valued local martingales.