Let $f(z) = z + g(z)$ where $g$ is holomorphic. Suppose that $|Img(z)| < 1$ for $z \in [−1−i,1−i]\cup[−1+i,1+i]$ and $|Reg(z)| < 1$ for $z \in [−1−i,−1 + i]\cup[1−i,1 + i]$. Show that $f$ has exactly one zero on the square $Q = \{x + iy \in \mathbb C : |x| < 1,|y| < 1\}$.
I have been trying to find the appropriate function to compare to $g(x)$ in order to apply the theorem, the most obvious candidate would be the function $h(z) = z$. I can show that this function is greater than $g(z)$ on the corners of the square but not anywhere else.
Any tips? Thanks!