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 + 1
2
| > 1
2
}⋂
{w : |w| < 1} .
3. Suppose that f is a holomorphic function in an open disk D, f is continuous in 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
1
P (z)
dz has the property that
I(r)
6= 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\{z0} for some z0
6= 0,
has an essential singularity at z0 and is continuous in {z : |z| ≤ |z0|}. 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(reiθ), 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) =
1
2π
∫ 2π
0
reiθ + z
reiθ − z
Re{f(reiθ)}dθ + i Im f(0) for |z| < r < R.
8. Suppose that f is an entire function, and for r > 0 let Mf (r) = sup{|f(z)| : |z| ≤ r}.
Assume that 0 < α < 1 and let
L(α) = lim
r→∞
Mf (αr)
Mf (r)
.
(a) Determine L(α) in the case f is a polynomial.
(b) Show that L(α) = 0 if f is not a polynomial.