Analytic functions with equal modulus differ by a unimodular constant

Let f and g be non-zero analytic functions on a region Ω. Assume \(| f ( z ) | = | g ( z ) |\) for all z in Ω. Show that \(f ( z ) = e ^ { i \theta } g ( z )\) in Ω for some \(0 \leq \theta < 2 \pi\) .