for
in the unit disk (Rogosinski 1934, p. 260; Osserman
2000). For
that is not a rotation, this follows by applying the
Schwarz-Pick lemma to , with its value at the origin
defined as .
If
extends continuously to a point with , , and exists, then the boundary Schwarz lemma gives
This bound follows from Unkelbach (1938, p. 741) and was independently obtained by Osserman (2000). In particular, unless is a rotation. Both estimates
are sharp, as shown by
for ,
with equality in the growth estimate for real and in the boundary estimate
at .
be analytic on the unit
disk, and assume that
for all 
and
.
and 
.
for some 
or if 
,
then 
is a rotation, i.e., 
for some complex constant 
with 
.
in the unit disk (Rogosinski 1934, p. 260; Osserman
2000). For 
that is not a rotation, this follows by applying the
Schwarz-Pick lemma to 
, with its value at the origin
defined as 
.
extends continuously to a point 
with 
, 
, and 
exists, then the boundary Schwarz lemma gives
unless 
is a rotation. Both estimates
are sharp, as shown by 
for 
,
with equality in the growth estimate for real 
and in the boundary estimate
at 
.
