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Classical and new loglog-theorems. (English) Zbl 1177.31001
Summary: We present a unified approach to the celebrated loglog-theorems of Carleman, Wolf, Levinson, Sjöberg, Matsaev on majorants of analytic functions. Moreover, we obtain stronger results by replacing the original pointwise bounds with integral ones. The main ingredient is a complete description for radial projections of harmonic measures of strictly star-shaped domains in the plane, which, in particular, explains where the loglog-conditions come from.

MSC:
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30D45 Normal functions of one complex variable, normal families
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