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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Summer 2018 • Question 6

Question

Math 6210 • Real Analysis Summer 2018 • Question 6

Let \(a_j\in \mathbb C\) satisfy \(|a_j|\le 1\) for all \(j\ge 2\). Show that \(f:B(0,\frac {1}{100})\to \mathbb C\) defined by \[ f(z)=z+\sum _{j=2}^{\infty }a_jz^j \] is injective.

Raw LaTeX

Let \(a_j\in\mathbb C\) satisfy \(|a_j|\le1\) for all \(j\ge2\). Show
that \(f:B(0,\frac{1}{100})\to\mathbb C\) defined by
\[
f(z)=z+\sum_{j=2}^{\infty}a_jz^j
\]
is injective.

Topic Tags

Automorphism Groups and Biholomorphisms