Vol. 85, No. 2, 1979

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The Schwarzian derivative and the Poincaré metric

Jacob Burbea

Vol. 85 (1979), No. 2, 345–354
Abstract

Let Ω0G be a plane region and let λΩ(z) be its Poincaré metric. Let EΩ be the complement of Ω and write α(ζ) = α(ζ : Ω) = {π1 EΩ |z ζ|4 (z)}12, where (z) = dxdy and ζ Ω. λΩ(z) = α(z : Ω) for all z Ω only when Ω is a disk less (possibly) a closed subset of inner capacity zero. Let ϕ be holomorphic and univalent in Ω and let Sϕ(z, ζ) = 6(2∕∂z∂ζ) × log(ϕ(z) ϕ(ζ))(z ζ). Here Sϕ(z,z) is the Schwarzian derivative of ϕ. We show

                            α2-(ζ) 1∕2
|Sϕ(z,ζ)| ≦ 6λΩ(z)λΩ(ζ)[1+ (1− λ2Ω (ζ)) ] : z,ζ ∈ Ω.

Mathematical Subject Classification 2000
Primary: 30C35
Milestones
Received: 31 October 1978
Revised: 19 March 1979
Published: 1 December 1979
Authors
Jacob Burbea