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Math Help - Question on power series coefficient of bounded function

  1. #1
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    Question on power series coefficient of bounded function

    I have a function f holomorphic on D(0;R).

    s, r such that (r < s < R)

    M(r) = sup { |f(z)| : z = r } (r < R)
    M(s) = sup { |f(z)| : z = s } (s < R)

    I was asked to show that M(r) <= M(s) and I have used the Maximum Modulus and the Identity theorems to accomplish that.

    The next part of question asks to show that M(r) r^(-n) >= M(s) s^(-n) when f is a polynomial of degree n 0 < r < s < R.

    I know that f(z) = Sum ( Cn (z - a)^n ) from 0 to inf

    and

    M(r) r^(-n) >= | An |
    M(s) r^(-n) >= | Bn |

    Could someone push me in the right direction?
    Thanks!
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  2. #2
    Super Member girdav's Avatar
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    Re: Question on power series coefficient of bounded function

    Use maximum modulus principle with Q(z):=z^nP(z^{-1}).
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