Application of Rouche's theorem gives two different answers? So I am supposed to find how many solutions the equation $z^7-5z^4+iz^2-2 = 0$ has in the region $|z|<1$. Here's the dilemma:
$|z^7-5z^4+iz^2|= |(-1)(-z^7+5z^4-iz^2)| = |-z^7+5z^4-iz^2| \geq -|z|^7+|5z^4-iz^2|\geq -|z|^7+5|z|^4-|z|^2 = -1+5-1 = 3 > 2$ (using the triangle inequality) on the circle $|z|=1$, which tells us that $f(z)$ has 7 solutions within the region.
However,
$|z^7+iz^2-2|\leq|z|^4+|iz|^2+|-2| \leq 1+1+2 = 4 \leq 5 = |-5z^4|$, which leads me to believe that there are 4 solutions within the region.
I am really confused! I am probably making a really silly mistake. Can someone please help me out? Thank you very much! 
 A: Well, the first method does not imply that the equation has $7$ solutions. More precisely, by Rouches theorem it has the same number of solutions as an upper dominant, which is $z^7-5z^4+iz^2.$ There nothing wrong with your second method, except that the last inequality has to be made strict and I would suggest to use in this particular case.
A: Wah, my bad. This was me being stupid. They both have 4 solutions (including multiplicity) in the region. Now it makes sense. I forgot initially to consider the fact that I am solely supposed to consider the solutions within the region $|z| <1$.
Using Wolfram Alpha, I discovered that in the region $|z|<1$ the factorization for $z^7−5z^4+iz^2 = 0$ gives us 3 solutions, two of multiplicity one, and one of multiplicity two. Meanwhile, in the same region, $−5z^4=0$ gives us one solution of multiplicity 4. Thus, we have including multiplicity, 4 solutions in the region for the equation $z^7−5z^4+iz^2−2=0$. Note that Rouche's theorem applied here does not give us the number of  distinct factors $z-a$ with $a\in\mathbb{C}$ such that $|a|<1$ but instead tells us $z^7−5z^4+iz^2−2= p(x)q(x)$ where $p(x)$ is of degree 3 and has all its roots in $|z|>1$ and $q(z)$ is of degree 4 and has all its roots in $|z|<1$.
(That is what it means to have 4 solutions including multiplicity.)
I would like to thank leshik for his help! 
