Makarov's principle for the Bloch unit ball

TL;DR

This paper explores Makarov's principle for Bloch functions, establishing bounds on key characteristics and improving existing estimates for the supremum of variance-related quantities within the Bloch unit ball.

Contribution

It demonstrates that universal bounds for variance-like quantities of Bloch functions coincide when taking the supremum over the Bloch unit ball and provides improved estimates over previous bounds.

Findings

Universal bounds for asymptotic variance, iterated logarithm constant, and spectrum second derivative agree for Bloch unit ball.

The supremum of these quantities is bounded above by 0.9, improving previous estimates.

The results refine the understanding of variance characteristics in Bloch functions.

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 $0.685^2 < \Sigma^2_{\mathcal B} \le 1$.