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Suppose $f:D(0,1)\longrightarrow \mathbb{C}$ is holomorphic, where $D(0,1)=\{z\in\mathbb{C}∣|z|<1\}$, and assume the maximum $|f(z)|\leq 2$. Estimate: $|f^{(3)}(i/3)|$.

I just don't understand how the solution had $R$ chosen as $R=\frac{2}{3}$

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    This is a repost of the same question which was answered. http://math.stackexchange.com/questions/218351/estimate-f3-i-3-using-cauchy/218368#2183682012-10-24

1 Answers 1

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$2/3$ is the minimum distance between $i/3$ and the unit circle. We have $|f^{(3)}(i/3)|=\left|\frac{1}{2\pi i}\oint \frac{f(z)}{(z-i/3)^4}\right|\le \frac{1}{2\pi}\oint \frac{|f(z)|}{|z-i/3|^4}|dz| \\ \le \frac{1}{2\pi}\oint \frac{2}{(2/3)^4}|dz|=\frac{3^4}{2^3}\frac{2\pi}{2\pi}$