I am not sure where to go with the following problem:
Do the functions $f(z) = e^z+z$ and $g(z)= ze^z+1$ have the same number of zeros in the strip $-\pi/2 < \text{Im}\, z< \pi/2$?
I tried letting $\gamma $ be the curve that traces the rectangle with vertices $(R + i\pi/2), ( -R + i \pi/2), (-R - i\pi/2), (R - i \pi/2)$ with the positive orientation, $R>1$ and then I wanted to apply Rouche's theorem to show $|f(z)+g(z)| < |f(z)| +|g(z)|$ on $\gamma$ but I couldn't get the inequality to work out.
Any suggestions?