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Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales
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- Journal:
- Canadian Mathematical Bulletin / Volume 55 / Issue 3 / 01 September 2012
- Published online by Cambridge University Press:
- 20 November 2018, pp. 597-610
- Print publication:
- 01 September 2012
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- Article
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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
$${{\left\| v \right\|}_{p}}\le {{C}_{p}}{{,}_{\infty }}{{\left\| u \right\|}_{\infty }},\,\,\,\,\,\,\,\,\,\,\,{{\left\| v \right\|}_{1}}\le {{C}_{1,p}}{{\left\| u \right\|}_{p}}.$$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.
