  The now canonical proof of Schwarz's Lemma appeared in a 1907 paper of
Carath\'eodory, who attributed it to Erhard Schmidt. Since then, Schwarz's
Lemma has acquired considerable fame, with multiple extensions and
generalizations. Much less known is that, in the same year 1907, Landau and
Toeplitz obtained a similar result where the diameter of the image set takes
over the role of the maximum modulus of the function. We give a streamlined
proof of this result and also extend it to include bounds on the growth of the
maximum modulus.
