Sharp LlogL inequalities for differentially subordinated martingales and harmonic functions

Abstract

Let $(x_n)$, $(y_n)$ be two martingales adapted to the same filtration $(\mathcal{F}_n)$ satisfying, with probability $1$, \[ |dx_n|\leq|dy_n|,\quad n=0, 1, 2, \ldots . \] For every $K>0,$ we determine the best constant $L=L(K)$ for which the inequality \[ \mathbb{E} |x_n| \leq K\mathbb{E} |y_n|\log|y_n|+L,\quad n=0, 1, 2, \ldots \] holds true. We also prove a similar inequality for harmonic functions.