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# Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales

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Published:2011-06-08
Printed: Sep 2012
• Adam Osękowski,
Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
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## Abstract

We determine the best constants $C_{p,\infty}$ and $C_{1,p}$, $1 < p < \infty$, for which the following holds. If $u$, $v$ are orthogonal harmonic functions on a Euclidean domain such that $v$ is differentially subordinate to $u$, then $$\|v\|_p \leq C_{p,\infty} \|u\|_\infty,\quad \|v\|_1 \leq C_{1,p} \|u\|_p.$$ In particular, the inequalities are still sharp for the conjugate harmonic functions on the unit disc of $\mathbb R^2$. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.
 Keywords: harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem
 MSC Classifications: 31B05 - Harmonic, subharmonic, superharmonic functions 60G44 - Martingales with continuous parameter 60G40 - Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

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