Analysis General Exam August 2009

Closed book, closed notes. Please pledge. In each problem, justify all assertions, show calculations, and identify those theorems that you invoke in your arguments.

Problem 1

Suppose that $f$ is a real-valued function on the real line $\mathbf{R}$ with $f$ absolutely continuous in the sense that it satisfies the fundamental theorem of calculus,

$$f(b)-f(a)=\int_{a}^{b} \frac{d f(x)}{d x} d x$$

for $a, b \in \mathbf{R}$, and for $d f / d x$ an $L^{1}(\mathbf{R})$-function, i.e., integrable over $\mathbf{R}$.
a) Use the definition of the Lebesgue integral and basic theorems of real analysis to show that given $\epsilon>0$, there exists a simple function $g$ of compact support so that $\|g-d f / d x\|<\epsilon$, with $\|\cdot\|$ the $L^{1}$ norm.
b) Prove that $f$ is uniformly continuous and that $\lim _{x \rightarrow \infty} f(x)$ exists.

Problem 2

Let $X$ be a measure space with measure $\mu$ and let $\left\{f_{n}\right\}$ be a sequence of real-valued measurable functions on $X$ and $f$ another measurable function so that

$$\mu\left\{x \in X:\left|f(x)-f_{n}(x)\right| \geq \frac{1}{2^{n}}\right\}<\frac{1}{2^{n}}$$

a) Show that the sequence $\left\{f_{n}\right\}$ converges to $f$ pointwise, a.e.
b) Suppose in addition that the functions of the above sequence are uniformly integrable in the sense that

$$\sup _{n} \int\left|f_{n}\right| d \mu \leq c$$

for some finite constant $c$. Show that $f$ is integrable.

Problem 3

Let $\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}$ be the unit disc in the complex plane. For $f$ analytic in $\mathbb{D}$ with power series

$$f(z)=\sum_{n \geq 0} a_{n} z^{n}$$

let

$$\|f\|_{2}=\left(\sum_{n \geq 0}\left|a_{n}\right|^{2}\right)^{1 / 2}$$

Let $\mathcal{F}$ be the collection of such analytic functions with finite $\|\cdot\|_{2}$-norm.
a) Show that

$$\frac{1}{2 \pi} \lim _{r \uparrow 1} \int_{\{z:|z|=r\}}|f(z)|^{2}|d z|=\|f\|_{2}^{2}$$

( $|d z|$ is differential arclength). Hint: Use polar coordinates.
b) Show that if $\left\{f_{k}\right\}$ is a Cauchy sequence in $\mathcal{F}$ with respect to the norm $\|\cdot\|_{2}$ defined above, then $\left\{f_{k}(z)\right\}$ converges uniformly for $z \in \mathbf{K}, \mathbf{K}$ any compact set contained in $\mathbb{D}$, and that the limiting function is analytic in $\mathbb{D}$.

Problem 4

Suppose that $P(z)$ is a polynomial of degree $n \geq 2$ with $n$ distinct zeros $z_{1}, z_{2}, \cdots, z_{n}$. Show that

$$\sum_{j=1}^{n} \frac{1}{P^{\prime}\left(z_{j}\right)}=0$$

Problem 5

Suppose that $f$ is analytic in a open set containing the upper half plane $\{z: \operatorname{Im} z \geq 0\}$, and suppose further that for some $M, a>0,|f(z)| \leq M /|z|^{a}$ for $|z|$ large. Show that for any $z$ with $\operatorname{Im} z>0$,

$$f(z)=\frac{1}{2 \pi i} \int_{-\infty}^{\infty} \frac{f(x)}{x-z} d x$$

Problem 6

True/False: Either prove that the statement is correct, or give a counterexample.

(a) If $f$ is analytic in the entire complex plane $\mathbb{C}$ and $\operatorname{Im} f \leq 0$ for all $z \in \mathbb{C}$, then $f$ is constant.
(b) If $f$ is analytic in the entire complex plane and bounded on the real axis, the $f$ is constant.
(c) If $f$ is analytic in the entire complex plane and $f(x+1)=f(x)$ for every real number $x$, then $f(z+1)=f(z)$ for every $z \in \mathbb{C}$.
(d) If $f$ is analytic in the entire complex plane with $|f(z)| \geq|z|$ for all $z \in \mathbb{C}$ and $f(1)=1$, then $f(z)=z$ for all $z \in \mathbb{C}$.

Problem 7

Suppose that $f$ is an analytic map of the unit disk $\mathbb{D}=\{z \in \mathbb{C}:|z|<1\}$ into itself that is not the identity function. Show that $f$ can have at most one fixed point in $\mathbb{D}$.