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.