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The sharp bound for the deformation of a disc under a hyperbolically convex map
Author(s) -
Barnard Roger W.,
Cole Leah,
Pearce Kent,
Williams G. Brock
Publication year - 2006
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611506015917
Subject(s) - mathematics , geodesic , schwarzian derivative , regular polygon , pure mathematics , lemma (botany) , bounded function , euclidean geometry , domain (mathematical analysis) , mathematical analysis , function (biology) , convex function , generalization , combinatorics , geometry , ecology , poaceae , evolutionary biology , biology
We complete the determination of how far convex maps can deform discs in each of the three classical geometries. The euclidean case was settled by Nehari in 1976, and the spherical case by Mejía and Pommerenke in 2000. We find the sharp bound on the Schwarzian derivative of a hyperbolically convex function and thus complete the hyperbolic case. This problem was first posed by Ma and Minda in a series of papers published in the 1980s. Mejía and Pommerenke then produced partial results and a conjecture as to the extremal function in 2000. Their function maps onto a domain bounded by two proper geodesic sides, a ‘hyperbolic strip’. Applying a generalization of the Julia variation and a critical Step Down Lemma, we show that there is an extremal function mapping onto a domain with at most two geodesic sides. We then verify using special function theory that, among the remaining candidates, the two‐sided domain of Mejía and Pommerenke is in fact extremal. This correlates nicely with the euclidean and spherically convex cases in which the extremal is known to be a map onto a two‐sided ‘strip’. 2000 Mathematics Subject Classification 30C70.