Reading time: about 9 minutes (1869 words).

1995 April 10

Instructions. Work as many of the problems as you can. Each solution should be clearly written on a separate sheet of paper.

  1. Let $f(z) = \sum a_n z^n$ be an entire function.
    (a) Suppose that $|f(z)| \leq A |z|^N + B$ for all $z\in \mathbb C$ where $A, B$ are finite constants. Show that $f$ is a polynomial of degree $N$ or less.
    (b) Suppose that $f$ satisfies the condition: $|f(z_n)|\rightarrow \infty$ whenever $|z_n| \rightarrow \infty$. Show that $f$ is a polynomial.

  2. (a) State a form of the Cauchy theorem.
    (b) State a converse of the Cauchy theorem.

  3. Let $f(z) = \sum_{n=0}^\infty a_n z^n$ be analytic and one-to-one on $|z|<1$. Suppose that $|f(z)|<1$ for all $|z|<1$.
    (a) Prove that $\sum_{n=1}^\infty n |a_n|^2\leq 1$.
    (b) Is the constant 1 the best possible?

  4. Let $u(z)$ be a nonconstant, real valued, harmonic function on $\mathbb C$. Prove there exists a sequence ${z_n}$ with $|z_n|\rightarrow \infty$ for which $u(z_n)\rightarrow 0$.

  5. Find an explicit conformal mapping of the semidisk $H = \{z : |z| < 1, \mathrm{Real}\ z > 0\}$ onto the unit disk.
    (cf. '89 Apr #3, '06 Nov #2).

  6. Suppose $f(z)$ is a holomorphic function on the unit disk which satisfies $|f(z)|<1$ for all $|z|<1$.
    (a) State the Schwarz lemma, as applied to $f$.
    (b) If $f(0)=\frac{1}{2}$, how large can $|f'(0)|$ be?
    (cf. '06 Nov #3)