Minimum Modulus Principle

Let  be analytic on a domain , and assume that  never vanishes. Then if there is a point  such that  for all , then  is constant.

Let  be a bounded domain, let  be a continuous function on the closed set  that is analytic on , and assume that  never vanishes on . Then the minimum value of  on  (which always exists) must occur on . In other words,

REFERENCES:

Krantz, S. G. "The Minimum Principle." §5.4.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 77, 1999.