# Original Proof of the Schwarz lemma

The classical Schwarz lemma from one-variable complex analysis states that a holomorphic map $$f : \Delta(r) \to \Delta(R)$$ between two disks in the complex plane such that $$f(0)=0$$ satisfies $$|f(z)| \leq \frac{R}{r} | z |$$.

Does anyone have access to the original paper (I assume is due to Hermann Schwarz) that presents this? I'd be interested in the original proof of the result.

According to Osserman The proof is given in Gesammelte Mathematische Abhandlungen. II, Springer-Verlag, Berlin, 1890. I've checked the reference, and used a translator on a number of the pages Osserman specifically references, but can't find the exact place where the Schwarz lemma is stated (and proved).

• If I remember correctly, it was never explicitly stated by Schwarz, and his proof used Cauchy's theorem. The modern statement and proof is due to Caratheodory. Aug 8 at 2:21
• Were you there when Schwarz said it (or not)? Aug 8 at 5:46