This talk is about a classical problem in complex analysis and geometric function theory: finding geometric conditions for functions to belong in spaces of holomorphic functions. In particular, we will talk about necessary and sufficient conditions for a conformal mapping of the unit disk to belong to Hardy or weighted Bergman spaces by studying the harmonic measure and the hyperbolic metric in the image region. Moreover, we will describe the Hardy number of conformal mappings in terms of the harmonic measure and the hyperbolic distance and give some applications in comb domains.
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