Preliminary Examination in Complex Analysis

January 1995

Let C be the set of complex numbers and let D = {z ∈ C : |z| < 1}.

1. Let U : D → D be harmonic and f : D → D be analytic. Prove or disprove the following:

(1) f ∘ U is harmonic.

(2) U ∘ f is harmonic.

2. Show that there exists an unbounded analytic function f on D such that

∫_{D} |f′(z)|^{2} dA(z) < +∞,

where dA is area measure on D.

3. Suppose f is analytic in D − \{0\} and unbounded near z = 0. If the function |z|^{\sqrt{2}}f(z)

is bounded at z = 0, show that

lim_{z→0} |z|^{\sqrt{2}}f(z) = 0 and lim_{z→0} |z|^{\sqrt{2}/2}f(z) = ∞.

4. Let X be the space of analytic functions f in D such that

∥f∥ = sup_{z∈D}(1 − |z|^{2})|f(z)| < +∞.

If {f_{n}} is a sequence of functions in X such that ∥f_{n} − f_{m}∥ → 0 as n, m → +∞, show that there exists a function f ∈ X such that ∥f_{n} − f∥ → 0 as n → +∞.

5. Let f be analytic in D. Show that

sup_{z∈D}(1 − |z|^{2})|f′(z)| ≤ sup_{z∈D} |f(z)|.

6. Suppose f is analytic in D. For z ∈ D and 0 < r < 1 − |z| let B(z, r) = {w ∈ D : |z − w| < r}. Show that

|f(z)|^{π} ≤ \frac{1}{πr^{2}} ∫_{B(z,r)} |f(w)|^{π} dA(w),

where dA is area measure on D.

7. Evaluate the integral

I = ∫_{|z|=π} \frac{sin z}{z cos z} dz.

8. Suppose {a_{n}} is a sequence in D − {0} with ∑(1 − |a_{n}|) < +∞. Show that

∏_{n=1}^{∞} \frac{|a_{n}|}{a_{n}} \frac{a_{n} − z}{1 − \overline{a}_{n}z}

converges (uniformly on compact sets) to an analytic function in D.