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.

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. 
(a) State a form of the Cauchy theorem.
(b) State a converse of the Cauchy theorem. 
Let $f(z) = \sum_{n=0}^\infty a_n z^n$ be analytic and onetoone 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? 
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 $u(z_n)\rightarrow 0$.

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). 
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)