Concave conformal mappings and pre-vertices of Schwarz-Christoffel mappings

Abstract

Normalized conformal mappings of the disk onto the exterior of a convex polygon are studied via a representation formula provided by Schwarz’s lemma. Some conditions on the pre-vertices for corresponding SchwarzChristoffel mappings are obtained. There is a connection to finite Blaschke products that characterizes the pre-vertices and leads to a curious property of Blaschke products themselves. 1. Concave conformal mappings A conformal, meromorphic function on the unit disk D is said to be a concave mapping if its image is the complement of a compact, convex set. Examples we want to consider are Schwarz-Christoffel mappings onto the exterior of convex polygons, including mappings onto the complement of finite rectilinear slits. Following standard practice, we call a set of points c1, c2, . . . , cn on the unit circle pre-vertices if they correspond to the vertices of a polygon, convex or not, under a conformal mapping of D onto the interior or exterior of the polygon. Locating the pre-vertices is the problem of accessory parameters for Schwarz-Christoffel mappings, and we obtain some information on the conditions the pre-vertices satisfy. If f has the form (1) f(z) = 1 z + b0 + b1z + b2z 2 + · · · , then a necessary and sufficient condition for f to be a concave mapping is (2) 1 + Re { z f ′′(z) f ′(z) } < 0 , |z| < 1 . The simple pole at the origin guarantees that zf ′′(z)/f ′(z) is analytic there, and (3) z f ′′(z) f ′(z) = −2− 2b1z − 6b2z + · · · . Let us examine the condition (2) in more detail, always assuming (1). If (2) holds, then there is an analytic function ω of D into itself, with ω(0) = 0, for which 1 + z f ′′(z) f ′(z) = ω(z) + 1 ω(z)− 1 . Received by the editors April 8, 2011. 2010 Mathematics Subject Classification. Primary 30C55; Secondary 30J10. The authors were supported in part by FONDECYT Grant #1110321. c ©2012 American Mathematical Society 3495 Licensed to Pontiffcia Universidade Catolica. Prepared on Fri Aug 22 12:21:52 EDT 2014 for download from IP 146.155.94.33. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3496 M. CHUAQUI, P. DUREN, AND B. OSGOOD Moreover, ω′(0) = 0, and hence φ(z) = ω(z) z2 is analytic in D with |φ(z)| ≤ 1 there. In terms of φ, (4) z f ′′(z) f ′(z) = 2 z2φ(z)− 1 or f ′′(z) f ′(z) = 2 ( zφ(z) z2φ(z)− 1 − 1 z ) . It is representation (4) that will allow us to deduce some information on pre-vertices. Before turning to this particular problem we have several general observations. From (4) we obtain the upper and lower bounds 2 1 + |z|2 ≤ ∣∣∣z f ′′(z) f ′(z) ∣∣∣ ≤ 2 1− |z|2 . Equality holds in either inequality at a point z0 = 0 only if |φ(z0)| = 1, in which case φ is a unimodular constant. Then up to a rotation and translation f(z) = z+1/z, so the image of D is the complement of a slit of length 4. Next, writing |z||φ(z)| ∣∣∣z f ′′(z) f ′(z) ∣∣∣ 2 = ∣∣∣2 + z f ′′(z) f ′(z) ∣∣∣ 2 from (4) and using |φ(z)| ≤ 1, we see that (2) may be replaced by the stronger inequality (5) 1 + Re { z f ′′(z) f ′(z) } ≤ − 4 (1− |z|) ∣∣∣z f ′′(z) f ′(z) ∣∣∣ 2 . As above, equality holds at a nonzero point only if φ is a unimodular constant, and so f maps to the complement of a slit. While the derivation of (5) used only |φ(z)| ≤ 1, an application of the invariant form of Schwarz’s lemma to (4) results in an inequality involving the Schwarzian derivative. The result is more complicated, but the case of equality is more interesting. For the Schwarzian derivative of f , Sf = ( f ′′ f ′ )′ − 1 2 ( f ′′ f ′ )2 ,