Number of roots of $z^7-4z^3-1$ in $|z|<1$

How many roots does the following polynomial have in the open disc $\abs{z} < 1$? \[ f(z) = z^7 - 4z^3 - 1 .\]

Solution

Big: $M(z) = -4z^3$. Small: $m(z) = z^7 - 1$. Then on $\abs{z} = 1$, \[ \abs{m(z)} = \abs{z^7-1} \leq \abs{z}^7 + 1 = 2 < 4 = \abs{-4z^3} ,\] so $f$ and $M$ have the same number of zeros: three.