Abstract
Let Ω be a hyperbolic region in the complex plane λΩ the density of the hyperbolic metric on Ω Set δΩ (z) = dist(z, ∂Ω); l/δΩ is called the quasihyperbolic density on Ω. Roughly speaking, we show that holomorphic functions cannot distinguish between λΩ and l/δΩ while meromorphic functions sometimes can. More precisely, for a holomorphic function f on Ω the quantities |f′ (z)|/(z)/λΩ (z) and |f′ (z)|δ Ω(z) are both either uniformly bounded on Ω (that is, fis a Bloch function) or unbounded. With the Euclidean derivative |f′| replaced by the spherical derivative f # = | f′|/(1+|f|2), Lehto and Virtanen have observed that the analogous result is generally false. However, we characterize those regions for which there exists a finite constant n = n(Ω) such that f #(z)δΩ(z) ≤ nf #(z)λΩ(z)≤ nf # (z)δΩ(z), z ∈ Ω, for any meromorphic function on Ω. In addition, we present another characterization of Bloch functions. A holomorphic function f on Ω is not a Bloch function if and only if there is a sequence {z n} 221En=1 zin Ω and a sequence {ρn} 221En=1 of positive numbers such that ρn/δΩ (zn) → 0 and f (zn + ρnς) — f(zn) →aς where |a| = 1, locally uniformly son C.
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© 1988 Springer-Verlag New York Inc.
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Minda, D. (1988). Bloch and normal functions on general planar regions. In: Drasin, D., Kra, I., Earle, C.J., Marden, A., Gehring, F.W. (eds) Holomorphic Functions and Moduli I. Mathematical Sciences Research Institute Publications, vol 10. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9602-4_8
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DOI: https://doi.org/10.1007/978-1-4613-9602-4_8
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