Real and Complex Analysis General Exam Fall 2007

Problem 1

Suppose that ( $X, \mathcal{A}, \mu$ ) is a measure space, with $\mathcal{A}$ a $\sigma$-algebra, $\mu$ a finite measure. Let $\mathcal{A}_{0} \subset \mathcal{A}$ be a sub-algebra of sets which generates $\mathcal{A}$; thus $\mathcal{A}$ is the smallest $\sigma$-algebra generated by $\mathcal{A}_{0}$.
Let $\mathcal{B} \subset \mathcal{A}$ be the collection of subsets of $X$ with the property that for every $\epsilon$ and $B \in \mathcal{B}$, there is an $A \in \mathcal{A}_{0}$, with

$$\mu(A \Delta B)<\epsilon .$$

(Here, $\Delta$ is the symmetric difference, $C \Delta D=(C-D) \cup(D-C)$.)
Show that $\mathcal{B}$ is a $\sigma$-algebra containing $\mathcal{A}_{0}$, in particular that it is an algebra, and that it is closed under countable (increasing) unions.

Hint: You may assume that (do not prove!):

$$\begin{aligned} \mu(C \Delta D) & =\mu\left(C^{c} \Delta D^{c}\right) \\ \mu\left(\left(D_{1} \cup D_{2}\right) \Delta\left(E_{1} \cup E_{2}\right)\right) & \leq \mu\left(D_{1} \Delta E_{1}\right)+\mu\left(D_{2} \Delta E_{2}\right) \\ |\mu(C \Delta E)-\mu(D \Delta E)| & \leq \mu(C \Delta D) \end{aligned}$$

Problem 2

(a) Let $f(x)$ be a real-valued differentiable function on a neighborhood of $[a, b]$, and assume that $\frac{d f}{d x}(a)<0$, and $\frac{d f}{d x}(b)>0$. Show that there exists a $c \in(a, b)$ such that $\frac{d f}{d x}(c)=0$.
(b) Suppose again that $\frac{d f}{d x}(a)<\frac{d f}{d x}(b)$ and that $\lambda$ satisfies $\frac{d f}{d x}(a)<\lambda<\frac{d f}{d x}(b)$. Show that there exists a $c \in(a, b)$ with $\frac{d f}{d x}(c)=\lambda$.

Problem 3

Let ( $X, \mu$ ) be a measure space.
(a) Show that if $g$ is a non-negative square-integrable function on $X$, then

$$\int_{\{x: g(x) \geq \lambda\}} g(x) d \mu(x) \leq \frac{1}{\lambda} \int_{X} g^{2}(x) d \mu(x)$$

Suppose now that ( $X, \mu$ ) is a finite measure space and let $\left\{f_{n}\right\}_{n=1,2, \ldots}$ be a sequence of non-negative functions which are both integrable and square-integrable, i.e., in $L^{1}(X) \cap L^{2}(X)$, that they converge pointwise to a function $f(x)$, and that the limits

$$\lim _{n \rightarrow \infty} \int f_{n}(x) d \mu(x)=L \text { and } \lim _{n \rightarrow \infty} \int f_{n}^{2}(x) d \mu(x)=M$$

exist.
(b) Show that

$$\lim _{n \rightarrow \infty} \int f_{n}(x) d \mu(x)=\int f(x) d \mu(x)$$

Hints: Why is

$$\int f(x) d \mu(x) \leq \liminf \int f_{n}(x) d \mu(x) ?$$

To show an inequality in the other direction: Define cut-off functions:

$$f_{n, \lambda}(x) \equiv\left\{\begin{array}{c} f_{n}(x), \quad f_{n}(x) \leq \lambda \\ \lambda \text { otherwise } \end{array}\right.$$

which converge pointwise to

$$f_{\lambda}(x) \equiv\left\{\begin{array}{c} f(x), \quad f(x) \leq \lambda \\ \lambda \text { otherwise } \end{array}\right.$$

Show that, given $\epsilon$, there is a $\lambda$ such that (again, $\mu$ is a finite measure and use part (a))

$$\begin{aligned} \limsup _{n} \int f_{n}(x) d \mu(x) & \leq \limsup _{n} \int f_{n, \lambda}(x) d \mu(x)+\epsilon \\ & \leq \int f_{\lambda}(x) d \mu(x)+\epsilon \end{aligned}$$

which is clearly

$$\leq \int f(x) d \mu(x)+\epsilon$$

Problem 4

Let $f$ be an entire function on the complex plane, and suppose there is a positive integer $N$ such that

$$\frac{f(z)}{z^{N}} \rightarrow 0 \text { as }|z| \rightarrow \infty$$

Find an upper bound on the number of zeros of $f$ (counting multiplicity) in the complex plane in terms of $N$. Is the bound sharp?

Problem 5

Let $\mathcal{H}$ be the Hilbert space of $L^{2}(D, \mu)$ functions on the unit $\operatorname{disc} D=\{z \in \mathcal{C}:|z|<1\}$, with $\mu$ two-dimensional measure, $d \mu=d x d y=r d r d \theta$ in polar coordinates, with $\|f\|_{L^{2}(D)}$ its $L^{2}$-norm. Suppose that $f \in \mathcal{H}$ is moreover analytic in $D$.
(a) Show that, for $0<r<1$

$$f(0)=\frac{1}{2 \pi} \int_{0}^{2 \pi} f\left(r e^{i \theta}\right) d \theta$$

and that for $0<r_{1}<1$,

$$f(0)=\frac{1}{\pi r_{1}^{2}} \int_{0}^{2 \pi} \int_{0}^{r_{1}} f\left(r e^{i \theta}\right) r d r d \theta$$

(b) Show that

$$|f(0)| \leq \frac{1}{\sqrt{\pi} r_{1}}\|f\|_{L^{2}(D)}$$

for all $f \in \mathcal{H}$ which are analytic in $D$.

Problem 6

Use the method of residues to evaluate

$$\int_{-\infty}^{\infty} \frac{x \sin x}{x^{2}+4} d x$$

Show all estimates.

Problem 7

Suppose that $f$ is analytic on $A=\{z \in \mathbb{C}: 1 \leq|z| \leq 3\}$, and assume that $|f(z)| \leq 1$ for $|z|=1$ and $|f(z)| \leq 9$ for $|z|=3$. Prove that $|f(2 i)| \leq 4$.