Talk:Schwarz lemma

I think that the phrase f:D-->D is wrong, because f transforms from open unit disk to closed unit dist (and not from open to open)

Just another math undergrad here... Thanks for the page - Schwarz lemma was a result I needed when studying Hyperbolic geometry. —Preceding unsigned comment added by 86.152.31.253 (talk) 21:32, 23 January 2008 (UTC)

Well this is just the hypothesis, so we can ask anything we like of f (of course that doesn't mean that such an f can exist, but for example f(z)=z, or even f(z)=0, clearly takes D to D and fixes 0). However, what would happen we did allow the codomain to be the closed unit disc? For it to be a "new" function (i.e. the image not be contained in D) there would need to be a z such that |z|<1 but |f(z)|=1. By the maximum modulus principle this says that f is a constant, and in particular |f(0)|=|f(z)|=1. Since we ask that f(0)=0 for Schwarz's lemma, this wouldn't count as one of the functions the lemma applies to. To put it another way, it doesn't matter either way whether you let the codomain be the open or closed disc, the set of functions that are included is the same (when you take into account the f(0)=0 requirement). The same applies to the Schwarz-Pick theorem.

By the way I know the previous comment is more than a year old, but this might help someone in future (especially since the article let the codomain be closed in a previous revision). Quietbritishjim (talk) 13:19, 12 July 2009 (UTC)

It got changed again so that it included both conditions simultaneously. I favor the $f:D\to D$ version, so I got rid of the other, but I also added a note at the bottom about the ambiguity in the theorem statement so hopefully this doesn't happen yet again. I also note Rudin's R&C A uses yet another condition, that $||f||_{\infty }\leq 1$, though this is so close to the $f:D\to {\bar {D}}$ version I already discussed that I'm ignoring it. 75.76.162.89 (talk) 09:09, 27 May 2012 (UTC)

Why do we need to write $|f(z)|\leq 1$ for all z? Because we already assumed co-domain as unit disc. — Preceding unsigned comment added by 103.27.8.45 (talk) 20:12, 1 May 2016 (UTC)