I’m trying to use Rouché’s theorem to find the number of roots of the function $f(z)=z^5+3z^4+9z^3+10$ on $\vert z\vert<2$. I know that the answer is $3$, but I have been unsuccessful in proving it. I’ve tried nearly every combination of the components of $f$ which also have $3$ roots, applying the theorem up to $3$ times, but to no avail. I’ve even resorted to trying unrelated polynomials, but there’s a lot of those! A hint would surely benefit my sanity.
Taking the geometric sequence $1,3,9$ in the leading coefficients into account, multiply with $(z-3)$ to get
which has the same number of roots inside the disk $|z|<2$. Now
on $|w|=1$ has clearly the third order term as the dominating one.