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Bulletin of the London Mathematical Society Advance Access originally published online on December 19, 2008
Bulletin of the London Mathematical Society 2009 41(1):27-35; doi:10.1112/blms/bdn098
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© 2008 London Mathematical Society

A reverse Denjoy theorem

P. C. Fenton

Department of Mathematics and Statistics
University of Otago
Dunedin
New Zealand
pfenton@maths.otago.ac.nz

John Rossi

Department of Mathematics
Virginia Tech
Blacksburg, VA 26061-0123
USA

Received 27 November 2007. Revision received 28 August 2008.

Suppose that C1 and C2 are two simple curves joining 0 to {infty}, non-intersecting in the finite plane except at 0 and enclosing a domain D which is such that, for all large r, Formula has measure at most 2{alpha}, where 0 < {alpha} < {pi}. Suppose also that u is a non-constant subharmonic function in the plane such that u(z) = B(|z|, u) for all large z isin C1 {cup} C2. Let AD(r, u) = inf { u(z):z isin D and | z | = r }. It is shown that if AD(r, u) = O(1) (or AD(r, u) = o(B(r, u))), then limr -> {infty} B(r, u)/r{pi}/2{alpha} > 0 (or limr->{infty} log B(r, u)/log r ≥ {pi}/2{alpha}).

2000 Mathematics Subject Classification 31A05.


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