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!:(