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Math Help - 2 Complex Analysis Questions

  1. #1
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    Exclamation 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

    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!
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  2. #2
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    Quote Originally Posted by musicmental85 View Post
    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 View Post
    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.
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