Midterm Math 534
Autumn 2008
Do as many problems as you can. I don’t expect anyone will be able to do them all. Complete
problems are worth more than several half completed problems. Leave some time at the end of the
hour to reread your solutions to eliminate errors. If you use a theorem with a name, please be sure
to give the name of the theorem you are using and to indicate that you’ve checked the hypotheses.
1. Suppose fis entire, r > 0, and f(0) = 1. Show there is a zwith |z|=rsuch that e−zf(z)∈
[1,∞).
2. Let fbe analytic in the plane, except for isolated singularities, and satisfy
|f(z)| ≤ C1
|z−1|2+C2
|z−2|
for all z6= 1,2. Prove that there are constants aand bso that
f(z) = a
(z−1)2+b
z−1+c
z−2.
3. Let fbe an entire function, uniformly bounded in the strip {0≤Rez < 1}and satisfying
f(z+ 1) = Cf (z),
where Cis a real constant. Show that there are constants aand bsuch that
f(z) = aebz .
4. Suppose Ω ⊂Cis a region such that 0 belongs to a bounded component of C\Ω. Prove that a
single-valued bran ch of log zcannot be defined in Ω.
5. Let nbe a positive integer. Find all functions fwith the following properties:
(i) fis continuous in |z| ≤ 2 and analytic in |z|<2;
(ii) f(j)(0) = 0 for 0 ≤j≤n;
(iii) |f(z)| ≤ 2n+1 for |z|= 2;
(iv) f(1) = 1.
