# 2 Complex Analysis Questions

• Nov 19th 2007, 02:49 PM
musicmental85
2 Complex Analysis Questions
Hi,
I'm trying to revise for an exam and came across these two question which I don't know how to do:confused:

a) Suppose f is analytic in the annulus A={z:1<=z<=3} if f(2)=1, |f(z)>=2 on |z|=1 and |f(z)|>=3 |z|=3, show that f must have a zero in A

b)Suppose a polynomial is bounded by 1 in D(0,1). Show that all its coefficients are bounded by 1.

For part a would the Maximum Modulus Theroem apply and how does one apply it?

For part b I'm not sure what theorem/lemma I'm supposed to be using? I've gone through my book but nothings ringing any bells.

I'd really appreciate a leg up, I need to figure out how to do these!
Thanks!:(
• Nov 20th 2007, 01:47 AM
Opalg
Quote:

Originally Posted by musicmental85
a) Suppose f is analytic in the annulus A={z:1<=z<=3} if f(2)=1, |f(z)>=2 on |z|=1 and |f(z)|>=3 |z|=3, show that f must have a zero in A

Suppose not. Then 1/f(z) is analytic in A, and takes a larger value at z=2 than it does anywhere on the boundary of A. That contradicts the maximum modulus principle.

Quote:

Originally Posted by musicmental85
b)Suppose a polynomial is bounded by 1 in D(0,1). Show that all its coefficients are bounded by 1.

Cauchy integral formula: if the polynomial is p(z) then the coefficient of z^k is $p^{(k)}(0)/k!$. By the Cauchy integral formula, $p^{(k)}(0) = \frac{k!}{2\pi i}\oint\frac{p(z)}{z^{k+1}}dz$, where the integral is taken round the unit circle.