Maximum modulus principle inside a closed contour

Let Ω be a simply connected open set and let γ be a simple closed contour in Ω and enclosing a bounded region U anticlockwise. Let \(f : \Omega \longrightarrow \mathbb { C }\) be a holomorphic function and \(| f ( z ) | \leq M\) for all \(z \in \gamma\) . Prove that \(| f ( z ) | \leq M\) for all \(z \in U\)