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?

  • $\begingroup$ Use the mean value theorem on $f'$. $\endgroup$ – copper.hat Apr 21 '17 at 6:11

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.