(Hi. This is my first question here.)

A well known result in complex analysis says that there is an $\varepsilon\gt 0$ such that if $f$ is holomorphic in (a neighborhood of) the closed disk ${\mathbb D}$ of radius 1, and $f'(0)=1$, then $f({\mathbb D})$ contains a disk of radius $\varepsilon$.

This is due to Landau and, accordingly, the largest possible $\varepsilon$ is called Landau's constant. The standard proof (see, for example, the book **Complex Variables** by Berenstein-Gay) gives $\varepsilon\ge1/16$.

As far as I understand, the best known bounds are $$ \frac 12\lt \varepsilon\le\frac{\Gamma(\frac13)\Gamma(\frac56)}{\Gamma(\frac16)}=0.54325\dots $$

However, I have been unable to locate any proofs of the first inequality, or any updated treatments of the second one (due to Rademacher). For all I know, current bounds may be better, or there may be a standard source to read about this.

Could you please give me some suggestions on where to look, or ideas on how to improve the $1/16$ bound, even if shorter of $1/2$?

(Many thanks!)