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Assume that $f(z)$ is holomorphic on $\{|z|<R\}$ and has Taylor expansion $f(z)=\sum\limits_{n=0}^{\infty}c_nz^n$.
a) Let $M(r)=\sup\limits_{|z|=r}|f(z)|$. Prove that $|c_n|\leq\dfrac{M(r)}{r^n}\quad (0<r<R).$
b) Prove that if one of the equations in $a)$ becomes equality, i.e $|c_k|=\frac{M(r)}{r^k}$ then $f(z)=c_kz^k$.

So far, I have done part a). I'm stuck at part b). Could someone help me?

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  • $\begingroup$ Please show your work and answer for part a. $\endgroup$
    – amWhy
    Apr 2 at 21:41
  • $\begingroup$ What can you say about the supremum of $|f(z) - c_k z^k|$? The idea seems to be that we have pushed all of $f$'s modulus into the $k$th term, so there shouldn't be any left in the terms of other degrees. $\endgroup$ Apr 2 at 22:05

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