Variants of the Schwarz lemma

Originally published at 狗和留美者不得入内. You can comment here or there.

Take some self map on the unit disk , . If , has a removable singularity at . On , , and with the maximum principle on , we derive everywhere. In particular, if anywhere, constancy by the maximum principle tells us that , where . with the removable singularity removed has , so again, by the maximum principle, means is a constant of modulus . Moreover, if is not an automorphism, we cannot have anywhere, so in that case, .