Preliminary Examination Complex Analysis June 1997

1. Suppose that f is an entire function and f(C) ∩ {w : Re w = 0} = ∅. Prove that f is constant.

2. Find a conformal mapping of the open unit disk onto the domain

Ω = \{w : |w + \frac{1}{2}| > \frac{1}{2}\} ⋂\{w : |w| < 1\} .

3. Suppose that f is a holomorphic function in an open disk D, f is continuous in \overline{D} and |f(z)| is constant for z ∈ ∂D. Prove that f is a rational function.

4. Determine all polynomials P such that I(r) = ∫_{|z|=r} \frac{1}{P (z)}dz has the property that I(r) ≠ 0 for all r > 0 for which I(r) is well-defined.

5. Give an example of a function f which is holomorphic in C\{z_{0}} for some z_{0} ≠ 0, has an essential singularity at z_{0} and is continuous in {z : |z| ≤ |z_{0}|}. Show that the function given actually has these properties.

6. Suppose that the function f is holomorphic in {z : |z| < R}, and for each r (0 < r < R) let L(r) denote the length of the curve w = f(re^{iθ}), 0 ≤ θ ≤ 2π. Show that L(r) ≥ 2πr|f′(0)| and determine all functions for which equality holds.

7. Suppose that the function f is holomorphic in {z : |z| < R} for some R > 0. Prove

that f(z) = \frac{1}{2π} ∫_{0}^{2π} \frac{re^{iθ} + z}{re^{iθ} − z} Re\{f(re^{iθ})\}dθ + i Im f(0) for |z| < r < R.

8. Suppose that f is an entire function, and for r > 0 let M_{f}(r) = sup{|f(z)| : |z| ≤ r}.

Assume that 0 < α < 1 and let

L(α) = lim_{r→∞} \frac{M_{f}(αr)}{M_{f}(r)} .

(a) Determine L(α) in the case f is a polynomial.

(b) Show that L(α) = 0 if f is not a polynomial.