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.