The integral means spectrum and the law of the iterated logarithm for conformal mappings

The present survey paper is described most important results about estimates of the integral means for derivatives of conformal maps of the unit disk into simply connected domains on the complex plane. Detailed description of the results connected with estimates of the integral means spectrum near the origin and also near the point $t=-2$ is given. Results connected with Makarov's law of the iterated logarithm are described. A connection between the integral means spectrum and the law of iterated logarithm is shown. On this basis metrical properties of harmonic measure on Jordan curves are improved.