Your argument does show there are zeros for .
I'm working on the following problem where I need to find the zeros using Rouche's theorem:
z^4 + 3z^3 + 6 inside |z| = 2
So far, I have the following:
I let f(z) = 3z^3 and g(z) = z^4 + 6.
I have |z^4 + 6| < |z|^4 + 6 = 22 < |f(z)| = 24.
so, |f(z)| > |g(z)| and according to the theorem, f(z) and f(z) + g(z) have the name number of zeros, counting multiplicities inside |z| = 2.
Is the correct answer f(z) has 3 zeros and f(z) + g(z) also have 3 zeros inside z = |2| = 2 ? Or is it the correct answer that there are no zeros?
Thanks a lot