Maximum Modulus Theorem

Maximum and Minimum Modulus Theorems

Given a closed contour on a path $C,$ in a region ${\mathcal {R}}$ of the complex plane where the function $f(z)$ is a non-constant analytic function, the modulus $|f(z)|$ attains both its maximum and minimum in the region bounded by $C$ on $C$.

This fact is useful for estimating contour integrals.