Illinois Journal of Mathematics

Sharp LlogL inequalities for differentially subordinated martingales and harmonic functions

Adam Oşekowski

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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.

Article information

Illinois J. Math., Volume 52, Number 3 (2008), 745-756.

First available in Project Euclid: 1 October 2009

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Mathematical Reviews number (MathSciNet)

Primary: 60G42: Martingales with discrete parameter 31B05: Harmonic, subharmonic, superharmonic functions


Oşekowski, Adam. Sharp LlogL inequalities for differentially subordinated martingales and harmonic functions. Illinois J. Math. 52 (2008), no. 3, 745--756. doi:10.1215/ijm/1254403712.

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