A First Course in Complex Analysis · Problem 16

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

Mathematics — Complex Analysis · maximum-modulus-principle

Problem

Give another proof of (a variant of) the Maximum-Modulus Theorem 8.2.4 via Corollary 8.1.12, as follows: Suppose

f

is holomorphic in a region containing

\overline{D}[a,r]

, and

|f(z)| \leq M

for

z \in C[a,r]

. Given a point

z_0 \in D[a,r]

, show (e.g., by Corollary 8.1.12) that there is a constant

c \in \mathbb{C}

such that

\left|f(z_0)^k\right| \leq c\, M^k.

Conclude that

|f(z_0)| \leq M