GENERAL EXAM - ANALYSIS

January, 2011

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

Problem 1

Let $\left\{f_{n}\right\}$ be a sequence of real-valued continuous functions on $[0,1]$ which is monotone non-increasing $f_{n+1}(x) \leq f_{n}(x)$ for all $x \in[0,1]$ and such that

$$\lim _{n \rightarrow \infty} f_{n}(x)=0$$

a) Prove that the convergence is uniform.
b) Show that, if instead $\left\{f_{n}\right\}$ is again a monotone sequence of continuous converging pointwise to a function $f$ which is however not continuous, then the convergence is not uniform.

Problem 2

Let $\left\{f_{n}\right\}$ be a sequence of real-valued Borel measurable functions on $\mathbb{R}$.
a) Show that

$$f(x) \equiv \sup _{n} f_{n}(x)$$

and

$$g(x) \equiv \limsup f_{n}(x)$$

are measuable.
b) Define the set $K$,

$$K=\left\{x: f_{n}(x) \in(0,1) \text { for infinitely many } \mathrm{n}^{\prime} \mathrm{s}\right\} .$$

Show that $K$ is a Borel measurable set.

Problem 3

Let $m$ be Lebesgue measure, and suppose that $f(x, y)$ is a Lebesgue measurable non-negative function on the plane $\mathbb{R}^{2}$ such that

$$F(\lambda, y)=m\{x: f(x, y) \geq \lambda\}$$

satisfies

$$\int_{0}^{\infty} \int_{\mathbb{R}} \lambda^{r} F(\lambda, y) d y d \lambda<\infty$$

for some $r \geq 0$.
Let

$$G(\lambda, x)=m\{y: f(x, y) \geq \lambda\}$$

a) Show that

$$\int_{0}^{\infty} \int_{\mathbb{R}} \lambda^{r} G(\lambda, x) d x d \lambda<\infty$$

b) Show that $f \in L^{r+1}\left(\mathbb{R}^{2}, d x d y\right)$, i.e.,

$$\int_{\mathbb{R}^{2}} f^{r+1}(x, y) d x d y<\infty$$

Show also that

$$m \times m\left\{(x, y) \in \mathbb{R}^{2}: f(x, y) \geq \lambda\right\} \leq \frac{c}{\lambda^{r+1}}$$

with $m \times m$ Lebesgue measure on the plane and with $c$ a finite constant.

Problem 4

Let $\left\{f_{n}\right\}$ be the sequence of functions defined on $[0,2 \pi]$ with

$$f_{n}(x)=\sum_{k=1}^{n} \frac{e^{i k x}}{k^{3 / 4}}$$

a) Show that $\left\{f_{n}\right\}$ converges in an $L^{2}([0,2 \pi], d x)$-sense, $n \rightarrow \infty$.
b) Show that $\left\{f_{n}\right\}$ converges in an $L^{1}[(0,2 \pi], d x)$-sense, $n \rightarrow \infty$.

Problem 5

Using residue methods, find

$$\int_{-\infty}^{\infty} \frac{\cos x}{e^{x}+e^{-x}} d x$$

by considering

$$\int_{\Gamma} \frac{e^{i z}}{e^{z}+e^{-z}} d z$$

where $\Gamma$ is the rectangle as shown with a suitably chosen value for the height.

Problem 6

Suppose that $f$ is analytic in an open connected set $\Omega$, and that all values of $f$ on $\Omega$ lie in the disk of radius $M>0$ centered at 0 . Prove that

$$(*)\left|f^{\prime}(z)\right| \leq \frac{M}{d(z)}$$

for all $z \in \Omega$, where $d(z)$ is the distance from $z$ to the boundary of $\Omega$. Then show that (*) can be used to prove Liouville's theorem.

Problem 7

Suppose $f$ is analytic in a set containing the closed unit disk $\overline{\mathbb{D}}=\{z:|z| \leq 1\}$ with $f(-\log 2)=0$ and $|f(z)| \leq\left|e^{z}\right|$ for all $z$ with $|z|=1$. How large can $|f(\log 2)|$ be? (Here, $\log z$ denotes the principal branch of the logarithm.)

Problem 8

a) Find the image of the unit disk $\mathbb{D}=\{z:|z|<1\}$ under the mapping

$$g(z)=\frac{z+1}{1-z}$$

b) Find the image of all straight lines through the point $z=1$ under this mapping.
c) Show that the function

$$f(z)=e^{-g(z)}$$

is bounded on the unit disk. Determine the limit of $f(z)$ as $z \rightarrow 1$ along any line segment lying within the unit disk. What is the limit as $z \rightarrow 1$ along the unit circle $\{z:|z|=1\}$ ?