Classical and new loglog-theorems

We present a unified approach to celebrated loglogloglog-theorems of Carleman, Wolf, Levinson, Sjöberg, Matsaev on majorants of analytic functions. Moreover, we obtain stronger results by replacing 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 loglogloglog-conditions come from.