Assume that $f:D\to C$ is univalent holomorphic from the unit disk into the complex plane so that $f(0)=f'(0)-1=0$. What is the precise constant $r$ so that $f(D)$ does not contain the disk $D(0,r)$. Is it $r=1$?
$\begingroup$
$\endgroup$
5
-
2$\begingroup$ Perhaps it follows at once from Schwarz lemma that $r=1$. $\endgroup$– MathArtCommented Feb 15, 2022 at 18:00
-
2$\begingroup$ The question is not very clear to me, but you may be looking for the Koebe 1/4 theorem. $\endgroup$– Emil JeřábekCommented Feb 15, 2022 at 18:12
-
$\begingroup$ @LukaThaler No, not really. For any such $f$, the minimum modulus of $z\notin f(D)$ is between $1/4$ and $1$; the upper bound indeed follows from Schwarz's lemma, as noted by the OP. $\endgroup$– Emil JeřábekCommented Feb 16, 2022 at 9:01
-
$\begingroup$ The question can be restated as: what is the (Euclidean) inradius of a domain with conformal radius $1$? $\endgroup$– Emil JeřábekCommented Feb 16, 2022 at 9:06
-
$\begingroup$ @EmilJeřábek you are absolutely right! It seems that I've overlooked the condition that maps considered here are univalent. $\endgroup$– Luka ThalerCommented Feb 17, 2022 at 10:56
Add a comment
|