Let $a_0,a_1,a_2,a_3 \in \mathbb C$ with $a_3\neq 0$, define $g: \mathbb R \rightarrow \mathbb C$ by $$g(t)=a_0e^{it}+a_1e^{2it}+a_2e^{3it}+a_3e^{4it},$$ show that there is a point $t\in \mathbb R$ such that $|g(t)|>|a_0|$
My attempt: By considering the holomorphic function $g: \mathbb C \rightarrow \mathbb C$ by $$g(z)=a_0e^{iz}+a_1e^{2iz}+a_2e^{3iz}+a_3e^{4iz},$$ Is there any scope to apply minimum modulus principle on the domain $\{z\in \mathbb C~:~|z|<|a_0|\}$?