Transcription of Open mapping theorem
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Open mapping theorem
SOLUTIONS TO COMPLEX PRELIM PROBLEMSC IHAN BAHRANIn this document, I will collect my solutions to some (not all, probably not even half)of the Complex Prelim problems that were asked in the past in theorems get used a lot. I will copy some of them from David C. Ullrich swonderful bookComplex Made mapping theoremThis is very useful in general. It s easy to forget the connectedness assumption, so Iwill state it (V) denotes the set of analytic maps from an open mapping open andf H(V). Also letWbe an open andconnected set contained inV. Thenf(W)is either a singleton (that is,fis constantonW) or open 2011, 7 and Fall 2010, that there is no one-to-one conformal mapof the punctured disk{z C: 0<|z|<1}onto the annulus{z C: 1<|z|<2}.LetDdenote the unit open disk andAthe described annulus. Suppose, to the contrary,that there existsf H(D {0}) which is injective andf(D {0}) =A.
SOLUTIONS TO COMPLEX PRELIM PROBLEMS 6 0 7!1. This will ensure C 1 maps to a line as well. The assignments 1 + i7!0 2i7!1 0 7!1 can be realized by the LFT ˚(z) = (z (1 + i))(2i 0)
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