Complex Functions Prelim, January 2011
Below Ddenotes the disk D={z∈C:|z|<1}.
In all cases the word “analytic” is used interchangeably with “holomorphic”.
1. (a) State and prove Schwarz lemma.
(b) Let fbe analytic on Dwith the property that f(0) = 1 and the real part of f
is positive on D. Prove that
|f(z)| ≤ 1 + |z|
1− |z|.
2. In each of the cases below, determine whether there exists an analytic 1-1 mapping
from Uonto the complex plane. If there is, write down an explicit formula for such
a mapping. Otherwise, prove that no such mapping exists.
(a) U=D
(b) U={z:|z−2|<2}\{z:|z−1| ≤ 1}.
3. Compute the following integral. Give full justification for your reasoning.
Z∞
0
x2+ 1
x4+ 1dx.
4. Suppose that f, ϕ are analytic in a domain containing D, and that fhas no zeros on
∂D. State and prove a formula for the following integral using the zeros of f.
1
2πi Z
∂D
f0
f(z)ϕ(z)dz.
5. Let fbe a non constant analytic function on a domain containing 0. Assume f(0) = 0.
Prove that for any δ > 0 there exists > 0 such that f(δD)⊃D.
6. Let Fbe the family of all functions fanalytic in Dsuch that
ZZD
|f(x−iy)|2dxdy < 1.
Prove that Fis a normal family.