Best Constants in Martingale Version of Rosenthal's Inequality

Abstract

The following generalization of Rosenthal's inequality was proved by Burkholder: $A^{-1}_p\{\|s(f)\|_p + \|d^\ast\|_p\} \leq \|f^\ast\|_p \leq B_p\{\|s(f)\|_p + \|d^\ast\|_p\},$ for all martingales $(f_n)$. It is known that $A_p$ grows like $\sqrt{p}$ as $p \rightarrow \infty$. In this paper we prove that the growth rate of $B_p$ as $p \rightarrow \infty$ is $p/\ln p$.