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.