Hyperbolic metric and membership of conformal maps in the Bergman space

Betsakos, Dimitrios; Karafyllia, Christina; Karamanlis, Nikolaos

doi:10.4153/s000843952000034x

Cited by 3 publications

(2 citation statements)

References 9 publications

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“…By applying different methods, the first author showed in [12] that f ∈ H p (D) if and only if (1.4) is true. Later, in [3] Betsakos and the current authors generalized this condition to weighted Bergman spaces. More specifically, they proved the following theorem.…”

Section: Introductionmentioning

confidence: 92%

Geometric characterizations for conformal mappings in weighted Bergman spaces

Karafyllia¹,

Karamanlis²

2021

Preprint

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We prove that a conformal mapping defined on the unit disk belongs to a weighted Bergman space if and only if certain integrals involving the harmonic measure converge. With the aid of this theorem, we give a geometric characterization of conformal mappings in Hardy or weighted Bergman spaces by studying Euclidean areas. Applying these results, we prove several consequences for such mappings that extend known results for Hardy spaces to weighted Bergman spaces. Moreover, we introduce a number which is the analogue of the Hardy number for weighted Bergman spaces. We derive various expressions for this number and hence we establish new results for the Hardy number and the relation between Hardy and weighted Bergman spaces.

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Section: Introductionmentioning

confidence: 92%

Geometric characterizations for conformal mappings in weighted Bergman spaces

Karafyllia¹,

Karamanlis²

2021

Preprint

Self Cite

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show abstract

“…In [7] the current author established another necessary and sufficient integral condition involving this time the hyperbolic distance d D (0, F r ). In [10] Betsakos, the author and Karamanlis generalized this condition to weighted Bergman spaces. We state all these results in the following theorem.…”

Section: Introductionmentioning

confidence: 99%