Prove that $z^5+(4/3)e^z \sin z-16iz+48$ has five zeroes in the disk $|z|<3$

Question

Prove that $z^5+(4/3)e^z \sin z-16iz+48$ has five zeroes in the disk $|z|<3$

I could prove it using Rouche's theorem if it werent for the second term. For other three terms, the modulus on the circle $|z|=3$ is easy to find. But how to estimate $e^z \sin z$ on this circle?

Answer 1

Convert trigonometric to exponential: $$e^z\sin z = e^z \frac1{2i}(e^{iz}-e^{-iz})$$ To estimate terms such as $e^{(1+i)z}$, proceed as follows: $$|e^{(1+i)z}| = e^{\operatorname{Re}((1+i)z)}\le e^{|(1+i)z|} = e^{\sqrt{2}|z|}$$ This will give you a decent bound on the circle $|z|=3$, enabling an application of Rouché's theorem.