I was reading about the maximum-minimum principle for harmonic functions in my lecture notes, and it was formulated like this:
Let $\phi$ be harmonic in a simply-connected domain $D$. If $\phi$ is bounded above or below by $M$, then if for some $z_0 \in D$, $\phi (z_0) = M$, then $\phi$ is constant in $D$.
But what does it mean for complex-valued function be bounded 'above'? The complex plane has many more 'directions' than the real line. Do they simply mean 'the real part is bounded above'?