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Let $f$ be a function analytic in the unit disk. The circle with center at the origin and radius $r$ is mapped by $f$ onto a curve whose length is denoted by $L(r)$. Prove the inequality $L(r) \geq 2r \pi |f^{\prime}(0)|$. Is it sharp?

I am trying to use the Schwarz lemma, since it is the only theorem in my book that talks about $|f^{\prime}(0)|$, but I cannot . Can you please help me?

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  • $\begingroup$ Use the mean value theorem on $f'$. $\endgroup$ – copper.hat Apr 21 '17 at 6:11
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The curve you're considering has a very concrete parametrization!

Namely, the curve $\gamma_r : [0,2\pi] \longrightarrow \mathbb C$ assigns $t\mapsto f(re^{it})$. The length of this curve is, by definition, $$ \int_0^{2\pi} |\gamma_r'(t)|dt = \int_0^{2\pi} r|f'(re^{it})|dt.$$

Consider now Cauchy's integral formula and the standard estimate.

This is all very independent of Schwarz's lemma.

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