# Number of zeros inside and outside unit disc

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.

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 $$|z|<4$$.
• Does the second part hold for radius less than $4$? – Math1 Apr 26 '20 at 14:24
• It holds when the radius is greater than $\sqrt{5+\sqrt{26}}\approx3.18$. – José Carlos Santos Apr 26 '20 at 14:28