The minimum principle in complex analysis, in my textbook, is stated like this:
Let $f: U\to \mathbb{C}$ be a non-constant analytical function, $f(z)\ne 0$, and U a conex of $\mathbb{C}$. Then the function: $g: U\to \mathbb{R}^+$, $g(z)=|f(z)|$ has no local minimums.
I've seen other variations of the principle, with varying degrees of formalism, and from what I gather, the idea is that if a function $f$ is analytical and has no zeros, then the minimum values of $|f(z)|$ must either be on the boundary of its domain or not exist. So in case the domain is an open set, since there is no boundary, the function can't have any mininums that aren't zeros at all.
The reason for this is that if the module of the function had a local minimum, then we would be able to define a disk/other compact set where the function was constant, and so, by the identity theorem, it would all be constant.
Is this analysis correct? I'm finding it hard to apply this in exercises, because it's a very strange idea to wrap ones mind around. Particularly the last explanation, with the identity theorem.
P.S.: If anyone knows of any online resources (video lectures, books, etc.) that offer more visual representation of complex analysis, that would be very helpful!