A First Course in Complex Analysis · Problem 12

Isolated Singularities and the Residue Theorem — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · maximum-modulus-principle

Problem

Prove the Minimum-Modulus Theorem (Corollary 8.2.6): Suppose

f

is holomorphic and nonconstant in a region

G

. Then

|f|

does not attain a weak relative minimum at a point

a

G

unless

f(a) = 0