PART I

Problem 1

Let $f$ be an entire function such that $|f(z)| \leq \log (10+|z|)$ for all $z$ in $\mathbb{C}$. Show that $f$ is a constant function.

Problem 2

Let $f$ be analytic in a neighborhood of the closed $\operatorname{disc} \overline{D\left(z_{0}, r\right)}$ and $0<s<r$. Show that

$$|f(z)| \leq \frac{1}{\pi(r-s)^{2}} \iint_{D\left(z_{0}, r\right)}|f(x+i y)| d x d y$$

for all $z$ in $\overline{D\left(z_{0}, s\right)}$.

Problem 3

Let $\Omega$ be an open set in $\mathbb{C}$ and $\mathcal{F}$ a family of analytic functions on $\Omega$ with the property that there exists a positive constant $M$ such that

$$\iint_{\Omega}|f(x+i y)| d x d y \leq M$$

for all $f$ in $\mathcal{F}$. Show that $\mathcal{F}$ is a normal family, i.e., there exists a sequence $\left\{f_{n}\right\} \subset \mathcal{F}$ such that $\left\{f_{n}\right\}$ converges uniformly on compacts in $\Omega$ (the inequality in problem 2. should be useful).

Problem 4

Show that

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

(it may prove convenient to consider the branch of the logarithm corresponding to the cut along the ’negative’ imaginary semi-axis).

Problem 5

Let

$$A=\{z \in \mathbb{C}: 0<|z|<1\} \text { and } B=\{z \in \mathbb{C}: 1<|z|<2\} .$$

Show that $A$ and $B$ are not conformally equivalent.

PART II

Problem 6

Let $\left\{E_{i}\right\}_{i=1}^{\infty}$ be a partition of a set $X$ (that is, $E_{i} \cap E_{j}=\emptyset$ and $\cup_{j=1}^{\infty} E_{j}=X$ ). For any $J \subseteq \mathbb{N}=\{1,2, \ldots\}$ let $E_{J}:=\cup_{i \in J} E_{i}$ with the convention that $E_{\emptyset}=\emptyset$.
(a) Show that $\mathcal{M}:=\left\{E_{J}\right\}_{J \subset \mathbb{N}}$ is the $\sigma$-algebra generated by $\left\{E_{i}\right\}_{i=1}^{\infty}$ (that is, the smallest $\sigma$-algebra containing $\left\{E_{i}\right\}_{i=1}^{\infty}$ ).
(b) Show that $f: X \rightarrow \mathbb{R}$ is $\mathcal{M}$-measurable iff $f$ is constant on each set $E_{i}$.

Problem 7

Let $f$ be a non-negative measurable function on $(0,1)$. Suppose there is a constant $c$ so that

$$\int_{0}^{1} f^{n}(x) d x=c \text { for all } n=1,2,3, \ldots$$

Show that there is a measurable set $A \subset(0,1)$ such that $f(x)=1_{A}(x)$ for a.e. $x \in(0,1)$.

Problem 8

Suppose $\mu$ and $\nu$ are finite (positive) measures on a measurable space ( $X, \mathcal{M}$ ). Show that there exists a measurable $f: X \rightarrow[0, \infty)$ such that for all $A \in \mathcal{M}$

$$\int_{A}(1-f) d \mu=\int_{A} f d \nu$$

Problem 9

Recall that, for a function $f: \mathbb{R} \rightarrow \mathbb{C}$, we say $f$ is Lipschitz if there exists $M \in(0, \infty)$ such that $|f(x)-f(y)| \leq M|x-y|$ for all $x, y \in \mathbb{R}$.
Show that $f: \mathbb{R} \rightarrow \mathbb{C}$ is Lipschitz iff $f$ is absolutely continuous and $\left|f^{\prime}\right| \leq M$.

Problem 10

Let $X=[-\pi, \pi]$ and $\lambda$ denote uniform measure on $X$, that is, $d \lambda(x)=\frac{1}{2 \pi} d x$. Then $H:=L^{2}(X, \lambda)$ equipped with the inner product

$$\langle u, v\rangle:=\int_{-\pi}^{\pi} u(x) \bar{v}(x) d \lambda(x)$$

is a Hilbert space, and, for $e_{k}(x):=e^{i k x},\left\{e_{k}\right\}_{k \in \mathbb{Z}}$ is an orthonormal basis of $H$. Suppose that $u \in C^{1}(\mathbb{R}, \mathbb{C})$ is $2 \pi$-periodic, that is,

$$u(x+2 \pi)=u(x) \text { for all } x \in \mathbb{R} .$$

(a) Show that $\left\langle u^{\prime}, e_{k}\right\rangle=i k\left\langle u, e_{k}\right\rangle$ for all $k \in \mathbb{Z}$, where $u^{\prime}(x)=\frac{d u}{d x}(x)$.
(b) Show that

$$s(x):=\sum_{k \in \mathbb{Z}}\left\langle u, e_{k}\right\rangle e^{i k x}$$

is convergent for all $x \in \mathbb{R}$ and that $s$ is continuous.
(c) Show that $u(x)=s(x)$ for all $x \in \mathbb{R}$.