Using Maximum Modulus Principle

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Homework Statement



Suppose p is a polynomial of degree n and |p(z)|≤M if |z|=1

Show that |p(z)|≤M|z|[itex]^{n}[/itex] if |z|≥1

Homework Equations



Maximum Modulus Principle: If f is a nonconstant analytic function on a domain D, then |f| can have no local maximum on D.

The Attempt at a Solution



Book says to apply the maximum modulus principle to [itex]\frac{p(z)}{z^{n}}[/itex] on domain |z|>1 but I am unsure why?
 

Answers and Replies

  • #2
Dick
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p(z)/z^n is 'analytic at infinity'. Has the book gone over what that means?
 
  • #3
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I believe f(z) is analytic at infinity if f(1/z) is analytic at z=0
 
  • #4
Dick
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I believe f(z) is analytic at infinity if f(1/z) is analytic at z=0

That's it. And p(z)/z^n is one of those functions, yes? It's just a polynomial in 1/z. If that's the case you can treat the function as analytic on the domain |z|>1.
 
  • #5
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so we can say that since p(z)/z^n is analytic at infinity then it is analytic on the domain |z|>1 so according to the maximum modulus principle, p(z)/z^n can have no local maximum on domain (|z|>1) right?
 
  • #6
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I'm thinking now I have to show that |p(z)/z^n|≤M if |z|≥1 is this correct?
 
  • #7
Dick
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so we can say that since p(z)/z^n is analytic at infinity then it is analytic on the domain |z|>1 so according to the maximum modulus principle, p(z)/z^n can have no local maximum on domain (|z|>1) right?

Yes, so |p(z)/z^n|<=M on |z|>1. Otherwise it would have a local maximum.
 
  • #8
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and M is just a real number? it is not specified anywhere... and what about |z|=1 since it says on |z|≥1 and we have looked at |z|>1
 
  • #9
Dick
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and M is just a real number? it is not specified anywhere... and what about |z|=1 since it says on |z|≥1 and we have looked at |z|>1

Of course M is real. Saying |p(z)|<=M doesn't make any sense if M is complex. The whole point is that if |f(z)|<=M on the boundary of an analytic domain then |f(z)|<=M on the interior of that domain. That's the maximum principle.
 
  • #10
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ok i thought so, just seemed like a weird variable for a real number.. but I think i understand it now, thank you so much
 

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