Analysis General Exam August 2016

PART I

Problem 1

An entire function $f$ is said to be of exponential type if there exist positive constants $c_{1}$ and $c_{2}$ such that

$$|f(z)| \leq c_{1} e^{c_{2}|z|}$$

for all $z$ in $\mathbb{C}$. Show that $f$ is of exponential type if and only $f^{\prime}$ is of exponential type.

Problem 2

Let $f$ be an entire function such that

$$|f(z)| \leq 2 e^{x} \text { for all } z=x+i y \text { such that }|z|>100 .$$

Show that $f$ is a multiple of the complex exponential function $e(z)=e^{z}$.

Problem 3

Let $\Omega$ be an open set containing the closed unit disc, and $\left\{f_{n}\right\}_{n=1}^{\infty}$ a sequence of analytic functions on $\Omega$ converging to a function $f$ uniformly on compacts (in $\Omega$ ). Suppose that the minimum of $|f|$ on the unit circle is a strictly positive number. Show that there exists a positive integer $n_{0}$ such that the functions $f_{n}$ have the same number of zeros in the open unit disc for all $n \geq n_{0}$.

Problem 4

Show that

$$\int_{-\infty}^{\infty} \frac{\sin x}{x} d x=\pi$$

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 ( $X, \mathcal{M}$ ) denote a measurable space and suppose $\mu, \nu$ are two positive measures on $\mathcal{M}$ such that $\mu \leq \nu$, that is, $\mu(A) \leq \nu(A)$ for all $A \in \mathcal{M}$. Show that

$$\int_{X} f d \mu \leq \int_{X} f d \nu$$

for all measurable $f: X \rightarrow[0, \infty]$.

Problem 7

For $p \in[1, \infty]$, let $L^{p}=L^{p}(X, \mathcal{M})$ where ( $X, \mathcal{M}$ ) is a measurable space.

Prove that, for $1 \leq p<q<r \leq \infty, L^{p} \cap L^{r}$ equipped with the norm $\|f\|:=\|f\|_{p}+\|f\|_{r}$ is a Banach space and the inclusion map $\iota: L^{p} \cap L^{r} \rightarrow L^{q}$ given by $\iota(f)=f$ is continuous with respect to the norm topologies.

Problem 8

Prove that

$$\lim _{n \rightarrow \infty} \int_{0}^{n}\left(1-\frac{x}{n}\right)^{n} d x=1$$

Problem 9

Suppose that $f$ is Lebesgue integrable on $(0,1)$ and $g:(0,1) \rightarrow \mathbb{R}$ is defined by

$$g(x):=\int_{x}^{1} \frac{f(t)}{t} d t$$

Prove that $g$ is integrable on $(0,1)$ and

$$\int_{0}^{1} g(x) d x=\int_{0}^{1} f(x) d x$$

Problem 10

Suppose $H$ is a Hilbert space and $M_{n} \subset H$ is an increasing sequence of closed subspaces. Let $M:=\cup_{n=1}^{\infty} M_{n}$ and $P_{n}=P_{M_{n}}$ and $P_{\bar{M}}$ be orthogonal projection onto $M_{n}$ and $\bar{M}$ (the closure of $M$ ) respectively. Show that

$$\lim _{n \rightarrow \infty} P_{n} x=P_{\bar{M}} x$$

for all $x \in H$.

Hint: First prove it for $x \in M^{\perp}$, then $x \in M$, and then $x \in \bar{M}$.