# 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, 2021 at 2:21
• Were you there when Schwarz said it (or not)? Aug 8, 2021 at 5:46

The following paragraph in "Julius and Julia: mastering the art of the Schwarz lemma" by Professor Harold P. Boas (Amer. Math. Monthly, 117 (2010), no. 9, pp. 770-785) might be relevant:

Now the special case of the lemma that Schwarz handled appears in the notes [41, pp. 109-111] from his lectures during 1869-70, but as far as I know, these notes did not see print until the publication of Schwarz's collected works in 1890...

Naturally, item 41 in the list of references of this paper is the second volume of Schwarz's Gesammelte Mathematische Abhandlungen (Springer, Berlin, 1890).

• Thank you very much! Aug 10, 2021 at 21:35