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Article overview
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Makarov's principle for the Bloch unit ball | Oleg Ivrii
; Ilgiz Kayumov
; | Date: |
1 May 2016 | Abstract: | Makarov’s principle relates three characteristics of Bloch functions that
resemble the variance of a Gaussian: asymptotic variance, the constant in
Makarov’s law of iterated logarithm and the second derivative of the integral
means spectrum at the origin. While these quantities need not be equal in
general, we show that the universal bounds agree if we take the supremum over
the Bloch unit ball. For the supremum (of either of these quantities), we give
the estimate $Sigma^2_{mathcal B} < min(0.9, Sigma^2)$, where $Sigma^2$ is
the analogous quantity associated to the unit ball in the $L^infty$ norm on
the Bloch space. This improves on the upper bound in Pommerenke’s estimate
$0.685^2 < Sigma^2_{mathcal B} le 1$. | Source: | arXiv, 1605.0246 | Services: | Forum | Review | PDF | Favorites |
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