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,
.