Complex Prelim, January 2006
is a complex polynomial. Show that
|p(eiθ)|2 dθ =
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
5. Find the Laurent expansion of the function
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 fn(z) is analytic in the unit disk |z| < 1. If
|fn(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)
6= 0, prove the existence of an analytic function
g(z) such that f(zn) = 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.