Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure

Abstract: We give affirmative answers to both parts of Question~2.6 posed by Grosse-Erdmann (2025) concerning Wiman--Valiron inequalities in the unit disk. For every unbounded analytic function in the disk, we establish the proposed iterated-logarithm inequalities outside exceptional sets of finite logarithmic measure. The corresponding power estimate is a corollary. Both conclusions follow from a variance bound for Khinchin families and a classical estimate for their largest atom. The multiplicative constants in the main inequalities can be chosen absolute. We also obtain a disk analogue of Rosenbloom's composition estimate, with an explicit boundary prefactor. The key step combines a boundary change of variable with a monotone auxiliary function whose derivative is exactly the variance of a rescaled member of the Khinchin family. A classical example shows that the leading logarithmic exponent $1/2$ cannot be decreased.