# Maximum Modulus Theorem

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