Let $f:C→P1$ be such that $f(z+1)=f(z+i)=f(z)$ for all z∈C. Let

$Γ=\{m+ni:m,n∈Z\}$. Show that if $f$ is holomorphic on $C∖Γ$, and $z⋅f(z)$ is bounded in a neighbourhood of $z=0$, then $f$ is constant.

I am stuck in the part how a condition at zero determines other point of the fundamental region. More specifically, how it prevents pole at other points on the boundary of the fundamental region (I assume that $f$ is holomorphic in the interior)