# Complex Analysis Roots Problem

I have some problem on finding roots and showing that they exist under certain conditions:

Problem: Prove that $e^z+z^3$ has no root in $\{z:|z|<3/4\}$ and has three roots in $\{z:|z|<2\}.$

I am preparing for my finals for complex analysis, and this is one of the review problems, but I cannot seem to solve it and its frustrating as I am stuck on this chapter complex roots and residues. I would appreciate if someone could show me how to solve this.

Hint: For the first part, consider the possible values of $|e^z|$ and $|z^3|$ in that disk. For the second part, use Rouche's Theorem.
• Sorry I am not sure, I still am quite stuck at showing $3$ roots exist in $|z|<2$. – Aurora Borealis Apr 18 '18 at 8:23
• @AuroraBorealis There are a bunch of examples on this site applying Rouche's Theorem to this type of problem; look at math.stackexchange.com/questions/773276/… and math.stackexchange.com/questions/1162675/… as examples. You want to express your function as a sum $f(z)+g(z)$, where you know how many zeroes $f$ has in $|z|<2$, and $|g(z)|<|f(z)|$ on the boundary $|z|=2$. – BallBoy Apr 18 '18 at 16:00