The Schwarzian derivative and the Poincaré metric

Jacob Burbea

Vol. 85 (1979), No. 2, 345–354

DOI: 10.2140/pjm.1979.85.345

Abstract

Let Ω∉0_G be a plane region and let λ_Ω(z) be its Poincaré metric. Let E_Ω be the complement of Ω and write α(ζ) = α(ζ : Ω) = {π⁻¹ ∫ _EΩ |z − ζ|⁻⁴ dσ(z)}^1∕2, where dσ(z) = dxdy and ζ ∈ Ω. λ_Ω(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 S_ϕ(z, ζ) = −6(∂²∕∂z∂ζ) × log(ϕ(z) −ϕ(ζ))∕(z −ζ). Here S_ϕ(z,z) is the Schwarzian derivative of ϕ. We show

Mathematical Subject Classification 2000

Primary: 30C35

Milestones

Received: 31 October 1978

Revised: 19 March 1979

Published: 1 December 1979