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Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales

  • Adam Osękowski (a1)
Abstract

We determine the best constants Cp , and C 1 ,p , 1 < p < ∞, 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

In particular, the inequalities are still sharp for the conjugate harmonic functions on the unit disc of ℝ2. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.

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References
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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