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1995 Apr

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)