Solution: ote that for $z\in \mathbb{C} $ \[ 0 \leq \vert f(z) \vert < \vert g(z) \vert \implies \vert g(z) \vert > 0 \implies g(z) \neq 0. \] COnsider the function \[ h(z) = \frac{f(z)}{g(z)}. \] Since $g(z)\neq 0$ and both functions are entire, the function $h$ is also an entire function. We have \[ \left\vert h(z) \right\vert = \left\vert \frac{f(z)}{g(z)} \right\vert < 1. \] Thus, $h$ is a bounded entire function. So by using the Liouville's theorem $h$ must be constant. Let $h(z) = \lambda $ for all $z \in \mathbb{C} $. Then \[ h(z) = \frac{f(z)}{g(z)} = \lambda \implies f(z) = \lambda g(z). \]