Number of zeros inside and outside unit disc

Question

Let $f(z) = 2z^{4}+5z^{2}$ and $g(z)=z^{4}+10z^{2}+1$. Prove that $f$ and $g$ have the same number of zeros inside the open unit disc as well as the same number of zeros outside the unit disc but inside the disc of radius $4$ centered at $0$.

Now, for the zeroes inside the unit disc we can apply Rouche's theorem we have $|g(z)-2f(z)|\le4<|2f(z)|\le14 $, when $|z|=1$ so $g$ and $f$ has the same number of zeroes inside the unit disc.

what about the other part of the question?

Thanks.

Answer 1

If $|z|=1$, then $|2z^4|<|5z^2|$ and $|z^4+1|<|10z^2|$. So, both $f$ and $g$ have $2$ zeros when $|z|<1$.

Now, if $|z|=4$, then $|5z^2|<|2z^4|$ and $|10z^2+1|<|z^4|$. So, both $f$ and $g$ have $4$ zeros when $|z|<4$.