I need to prove that the equation $z - \varepsilon \sin z = a$ has only one solution for small $\varepsilon$. The problem doesn't state anything specifically but I assume that $a$ is an arbitrary complex number.
I've tried using Rouche's theorem on circles with arbitrarily large diameters and with center in the origin of the coordinate system where I take $z - a$ as the dominating function and $\varepsilon \sin z$ as the smaller function, but I achieved no success. I would be very happy if someone could help me with this task.