Rouche's Theorem in Complex Analysis

Question

I am trying to find how many zeros of the given function f are located in the unit disk $|z|<1$:

$f(z) = e^z - 4z^n + 1$, where $n=1,2,3,...$

Using Rouche's Theorem, which states that if both $f$ and $g$ are holomorphic inside and on some closed contour $C$, and $|f(z)| > |g(z)|$ for all $z \in C$, then $f$ and $f+g$ have the same number of zeros inside of $C$. I started by letting $f(z) = -4z^n+1$ and $g(z) = e^z$. My problem is how to figure the answer out using Rouche's Theorem.

Answer 1

On $|z|=1$ $$ |e^z|=|e^{\cos x}e^{i\sin x}|=e^{\cos x}\leqslant e^1<3=|-4z^n|-1\leqslant |-4z^n+1| $$ So $e^z-4z^n+1$ has same zeros as $-4z^n+1$, which is $n$ in $|z|<1$.