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 Mat. Sb., 2017, Volume 208, Number 3, Pages 96–110 (Mi msb8727)

Makarov's principle for the Bloch unit ball

O. V. Ivriia, I. R. Kayumovb

a California Institute of Technology, Pasadena, CA, USA
b Kazan (Volga Region) Federal University

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 $\Sigma^2_{\mathscr B} < \min(0.9, \Sigma^2)$, 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 $0.685^2 < \Sigma^2_{\mathscr B} \le 1$.
Bibliography: 23 titles.

Keywords: Bloch space, law of the iterated logarithm, integral means spectrum, Bergman projection.

 Funding Agency Grant Number Academy of Finland 271983273458 Russian Foundation for Basic Research 14-01-00351-à15-41-02433-ð_ïîâîëæüå_à O. V. Ivrii's research was supported by the Academy of Finland (grants nos. 271983 and 273458). I. R. Kayumov's research was supported by the Russian Foundation for Basic Research (grant no. 14-01-00351_a) and by joint grant no. 15-41-02433-ð_ïîâîëæüå_a of the Russian Foundation for Basic Research and the government of the Republic of Tatarstan.

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DOI: https://doi.org/10.4213/sm8727

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English version:
Sbornik: Mathematics, 2017, 208:3, 399–412

Bibliographic databases:

UDC: 517.546.12+517.547.5
MSC: 30H30

Citation: O. V. Ivrii, I. R. Kayumov, “Makarov's principle for the Bloch unit ball”, Mat. Sb., 208:3 (2017), 96–110; Sb. Math., 208:3 (2017), 399–412

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8727
• https://doi.org/10.4213/sm8727
• http://mi.mathnet.ru/eng/msb/v208/i3/p96

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. O. Ivrii, “On Makarov's principle in conformal mapping”, Int. Math. Res. Not. IMRN, 2019, no. 5, 1543–1567
2. I. R. Kayumov, K.-J. Wirths, “On the sum of squares of the coefficients of Bloch functions”, Monatsh. Math., 190:1 (2019), 123–135
3. I. R. Kayumov, K.-J. Wirths, “Coefficient inequalities for Bloch functions”, Lobachevskii J. Math., 40:9 (2019), 1319–1323
4. I. R. Kayumov, K.-J. Wirths, “Coefficients problems for Bloch functions”, Anal. Math. Phys., 9:3 (2019), 1069–1085
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