I want to show that the equation $az^{3}-z+b=e^{-z}(z+2)$, where $a>0$ and $b>2$, has two roots in the right half-plane $\mathrm{Re}\;z\geq 0$. This is a problem in using Rouche's theorem but I am unable to get the right estimates.
I tried taking a semicircular contour in the right half-plane with its arc going from $-iR$ to $iR$ and then going down the imaginary axis. To apply Rouche's theorem, I need holomorphic functions $f,g$ such that $|g(z)|<|f(z)|$ for $z$ on the semicircle. I took $f(z)=az^{3}-z+b$ and $g(z)=e^{-z}(z+2)$. Things seem to be fine on the arc but I ran into problems on the imaginary axis.