Bounded Schwarzian and Two-Point Distortion
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Bounded Schwarzian and Two-Point Distortion W. Ma · D. Mejia · D. Minda
Received: 21 November 2012 / Revised: 18 October 2013 / Accepted: 19 October 2013 / Published online: 21 November 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract The Schwarzian derivative of a locally injective holomorphic function f is 2 S f = f / f − (3/2) f / f . As is well-known, S f = 0 if and only if f is a Möbius transformation. Intuitively, if a locally injective holomorphic function has a small Schwarzian derivative, then it should behave roughly like a Möbius transformation. Two quantitative results of this type are established. First, if |S f (z)| ≤ 2t, z ∈ , on a convex region , then sharp upper and lower two-point distortion bounds on | f (a) − f (b)| for a, b ∈ are given. The upper √ bound is valid for all a, b ∈ while the lower bound is valid for |a −b| < π/ t. For t = 0 the bounds are the familiar identity | f (a) − f (b)| = |a − b| | f (a)|| f (b)| for Möbius transformations. These upper and lower two-point distortion theorems characterize locally injective holomorphic functions with bounded Schwarzian derivative. Second, if is a convex
Communicated by Peter Duren. This work was initiated while the authors were participants in a Taft Research Seminar organized by David Herron at the University of Cincinnati. The first author gratefully acknowledges partial support from the Taft Research Center. The first two authors also thank the Department of Mathematical Sciences at University of Cincinnati for its hospitality during their visits. The second author was also partially supported by COLCIENCIAS, Project 1118–52128160. W. Ma School of Integrated Studies, Pennsylvania College of Technology, Williamsport, PA 17701, USA e-mail: [email protected] D. Mejia Escuela de Matemáticas, Bloque 43, Universidad Nacional, Calle 59A No. 63–20, Medellín, Colombia e-mail: [email protected] D. Minda (B) Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA e-mail: [email protected]
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region with diameter D and |S f (z)| ≤ 2t < π 2 /D 2 for z ∈ , then f is K t (D)-quasiMöbius, where the constant depends only on t and D. This means that 1/K t (D) ≤ | f (a), f (b), f (c), f (d)|/|a, b, c, d| ≤ K t (D) for all distinct a, b, c, d ∈ , where |a, b, c, d| denotes the absolute cross-ratio. Keywords Schwarzian derivative · Locally injective holomorphic functions · Two-point distortion · Absolute cross-ratio distortion Mathematics Subject Classification (2000)
Primary 30C55; Secondary 30C45
1 Introduction If f is a Möbius transformation, then it is straightforward to verify that for all a, b ∈ C | f (a) − f (b)| = |a − b| | f (a)|| f (b)|,
(1.1)
provided f (a), f (b) ∈ C. This identity implies that Möbius transformations preserve absolute cross-ratios, where the absolute cross-ratio is given by |a, b, c, d| =
|a − c||b − d| . |a − d||b − c|
The Schwarzian derivative of a locally injective meromorphic function f is f (z) 3 S f (z) = − f (z) 2
f
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