Geometric characterizations for conformal mappings in weighted Bergman spaces

Extremal Distance and Conformal Mappings in Hardy and Bergman Spaces

“…In [14,15] Poggi-Corradini gave a necessary and sufficient integral condition for f to belong to H p (D) in terms of the harmonic measure ω D (0, F r ). The current author and Karamanlis extended this condition to weighted Bergman spaces in [11]. In [7] the current author established another necessary and sufficient integral condition involving this time the hyperbolic distance d D (0, F r ).…”

“…where dA denotes the Lebesgue area measure on D. For the theory of Bergman spaces, see [4]. As an analogue of the Hardy number for weighted Bergman spaces, the current author and Karamanlis introduced in [11] the Bergman number as follows. If f is a conformal mapping of D, the Bergman number of f is defined by…”

“…They then proved that b(f ) = h(f ). See [11,12]. A well-studied problem in geometric function theory is to find geometric conditions for a conformal mapping f of D to belong to some space H p (D) or A p α (D) by studying the image region f (D).…”

“…A well-studied problem in geometric function theory is to find geometric conditions for a conformal mapping f of D to belong to some space H p (D) or A p α (D) by studying the image region f (D). See, for example, [7,11,15] and references therein. Some of these conditions that have been proved recently involve conformal invariants such as harmonic measure and hyperbolic distance.…”