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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Summer 2023 • Question 6

Question

Math 6220 • Complex Analysis Summer 2023 • Question 6

Suppose a sequence of holomorphic functions \(f_n:\mathbb {D}\to \mathbb {C}\) converges pointwise to a function \(f:\mathbb {D}\to \mathbb {C}\), and that all \(f_n\) are uniformly bounded. Show that convergence is uniform on every disk

\[ D_r=\{z\in \mathbb {C}\mid |z|\le r\} \]

for \(r<1\).

Raw LaTeX

Suppose a sequence of holomorphic functions \(f_n:\mathbb{D}\to\mathbb{C}\) converges
pointwise to a function \(f:\mathbb{D}\to\mathbb{C}\), and that all \(f_n\) are uniformly bounded.
Show that convergence is uniform on every disk

\[
D_r=\{z\in\mathbb{C}\mid |z|\le r\}
\]

for \(r<1\).

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Uniform Convergence and Hurwitz's Theorem Zeros of Holomorphic Functions and the Identity Theorem