Rouché theorem : Let $D \subset \mathbb{C}$ a domain and $f,g: D \to \mathbb{C}$ two holomorphic functions in $D$. Let $C$ a closed path contained in the interior of $D$. If $|f(z)+g(z)| < |f(z)|+|g(z)|$, $\forall z \in \mathbb{C}$, then $f$ and $g$ have the same number of zeroes in the interior of $C$.
Question : Show that if $a>e$, the equation $az^n=e^z$ admit $n$ roots in the unit disk.
So I've taken $f(z)=-az^n+e^z$ and $g(z)= az^n$ in using the closed path $C : |z|=1$. I obtained $|-az^n+e^z+az^n|= |e^z| \leq |-az^n+e^z|+|az^n|$, $\forall z \in C$.
How could I prove that $|e^z| \not= |-az^n+e^z|+|az^n|$?
Hint. On $\{\lvert z\rvert=1\}$ we have $$ \lvert\mathrm{e}^z\rvert\le \mathrm{e}<a\le \lvert az^n-\mathrm{e}^z\rvert+\lvert az^n\rvert. $$