Complex Prelim, January 2006

1. Suppose

p(z) = ∑_{n=0}^{N} a_{n}z^{n}

is a complex polynomial. Show that

\frac{1}{2π} ∫_{0}^{2π} |p(e^{iθ})|^{2} dθ = ∑_{n=0}^{N} |a_{n}|^{2}.

2. If u is a harmonic function defined on the complex plane and f is entire, show that u ∘ f is harmonic.

3. Construct a conformal mapping from the first quadrant of the complex plane onto the horizontal strip |y| < 1.

4. If f(z) is an entire function and its real part is bounded from below, show that f must be constant.

5. Find the Laurent expansion of the function

f(z) = \frac{2}{z(z − 1)(z − 2)}

in the annulus 1 < |z| < 2.

6. Suppose f(z) is entire and p(z) is a polynomial. If |f(z)| ≤ |p(z)| for all z, show that there exists a constant c such that f(z) = cp(z).

7. Characterize all anaytic functions f(z) in |z| < 1 such that |f(z)| ≤ | sin(1/z)| for all 0 < |z| < 1.

8. Suppose each f_{n}(z) is analytic in the unit disk |z| < 1. If ∑ |f_{n}(z)| converges uniformly for |z| < 1, show that ∑ |f′_{n}(z)| converges uniformly for |z| ≤ r, where r ∈ (0, 1).

9. If f(z) is analytic in |z| < 1 and f′(0) ≠ 0, prove the existence of an analytic function g(z) such that f(z^{n}) = f(0) + g(z)^{n} in a neighborhood of the origin.

10. If f(z) is analytic in |z| < 1 and |f(z)| ≤ 1 for all |z| < 1, show that (1−|z|^{2})|f′(z)| ≤ 1 for all |z| < 1.