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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Fall 2021 • Question 4

Question

Math 6220 • Complex Analysis Fall 2021 • Question 4

Show that if \(D\subsetneq B(0,1)\) is simply connected and \(0\in D\) then there exists an injective, holomorphic function \(f:D\to B(0,1)\) so that \(f'(0)>1\). -Note that this is part of the proof of the Riemann mapping theorem, so please do not use the Riemann mapping theorem in the proof.

Raw LaTeX

Show that if \(D\subsetneq B(0,1)\) is simply connected and \(0\in D\) then there exists an
injective, holomorphic function \(f:D\to B(0,1)\) so that \(f'(0)>1\).
-Note that this is part of the proof of the Riemann mapping theorem, so
please do not use the Riemann mapping theorem in the proof.

Topic Tags

Automorphism Groups and Biholomorphisms Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Riemann Mapping Theorem and Simply Connected Domains