Complex Analysis General Exam, August 2020

General guidelines: In order to receive full credit, a detailed argument (rather than a sketch of the proof) is needed. Whenever applying one of the standard theorems, please indicate that clearly.

Pledge:

Problem 1

Let $f$ be an entire function. Assume that

$$\max _{|z|=r}|f(z)| \leq 10 \log r$$

for $r \geq 100$. Shown that $f$ is constant.

Problem 2

Let $f$ be holomorphic on $\Omega=\{0<|z|<1\}$, and suppose that
(i) $|f(z)|>1$ for all $z \in \Omega$,
(ii) there exists a sequence $\left\{z_{k}\right\}_{k=1}^{\infty} \subset \Omega, z_{k} \rightarrow 0, k \rightarrow \infty$ such that $\left|f\left(z_{k}\right)\right| \leq 10$ for $k \geq 100$.

Is 0 a removable singularity?

Problem 3

Compute

$$\int_{0}^{\infty} \frac{\log x}{x^{2}+1} d x$$

via complex integration. Show all estimates.

Problem 4

Let $f: \mathrm{D}=\{|z|<1\} \rightarrow \mathbb{C}$ be holomorphic. Assume that $f(0)=0$ and $|\operatorname{Re} f(z)|<1$, for all $z \in \mathrm{D}$.
Show that

$$\left|f^{\prime}(0)\right| \leq \frac{4}{\pi}$$

Hint: A function $g(w)=\frac{e^{i \pi \frac{w}{2}}-1}{e^{i \pi \frac{w}{2}}+1}$ mapping $\{|\operatorname{Re} w|<1\}$ conformally to D might prove useful.

Problem 5

Let $\mathcal{F}$ be a family of entire functions with the property that for any circle $C$, there exists a constant $M_{C}>0$ such that

$$\sup _{f \in \mathcal{F}} \max _{z \in C}|f(z)| \leq M_{C}$$

(i) show that there exists a sequence $\left\{f_{n}\right\}_{n=1}^{\infty} \subset \mathcal{F}$ converging to an entire function $g$ uniformly on compacts in $\mathbb{C}$
(ii) suppose that $f_{n}(z) \neq 7$ for $n \geq 77$ and all $z \in\{|z|<1\}$. Can $g\left(\frac{1+i}{2}\right)=7$ ?