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Abstract

We determine the optimal constants $C_{p,q}$ in the moment inequalities $$ \|g\|_p \leq C_{p,q} \|f\|_q,\quad\ 1\leq p< q< \infty, $$ where $f=(f_n)$, $g=(g_n)$ are two martingales, adapted to the same filtration, satisfying $$ |dg_n|\leq |df_n|,\quad\ n=0,1,2,\ldots, $$ with probability $1$. Furthermore, we establish related sharp estimates $$ \|g\|_1 \leq \sup_n {\mathbb E} {\mit\Phi}(|f_n|)+L({\mit\Phi}),$$ where ${\mit\Phi}$ is an increasing convex function satisfying certain growth conditions and $L({\mit\Phi})$ depends only on ${\mit\Phi}$.