The law of the iterated logarithm for locally univalent functions

Abstract

In this paper we prove a sharp version of the Makarov law of the iterated logarithm. In particular, we show that the constant in the right side of this law depends on an asymptotic behaviour of the integral means of the derivative of an analytic function. Also, we establish that this constant is equal to the asymptotic variance for some domains with fractal type boundaries.