Maximum Modulus Principle
Let 
be a domain, and let 
be an analytic function on 
. Then if there is a point 
such that 
for all 
, then 
is constant. The following slightly sharper version can also be formulated. Let 
be a domain, and let 
be an analytic function on 
. Then if there is a point 
at which 
has a local maximum, then 
is constant.
Furthermore, let 
be a bounded domain, and let 
be a continuous function on the closed set 
that is analytic on 
. Then the maximum value of 
on 
(which always exists) occurs on the boundary 
. In other words,
The maximum modulus theorem is not always true on an unbounded domain.
REFERENCES:
Krantz, S. G. "The Maximum Modulus Principle" and "Boundary Maximum Modulus Theorem." §5.4.1 and 5.4.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 76-77, 1999.
