Minimum Modulus principle

Question

i would like to clarify for the theorem of minimum modulus, if $f(z) = 0$ for some $z$ in the region of analyticity, then our minimum modulus will occur at that point $z$ where $f(z) = 0$? Because the theorem always claim that $f(z)$ is non vanishing throughout, so i got a bit confused to what will happen if $f(z) =0$

Answer 1

Exactly as I had thought. So the minimum modulus principle does not apply to $z^2 - z$. Instead, the question is elementary ,and to find the minimum moduli you simply have to find the roots of $z^2 - z$ over the given domain, which occur when $z =0, z=1$. This is simple arithmetic, not requiring the min modulus principle.

What the maximum modulus principle will tell you (and it's applicable) is that since $z^2 - z$ is not constant, it's modulus must be taking a maximum on the boundary, so the advantage we get from MMT : it's enough to search on the boundary if you want the maximum, rather than search the whole domain.

A search on the boundary yields that $z=-1$ satisfies $|z^2 - z| = 2$, which is the largest it can be, since $|z^2-z| \leq |z^2| + |z| \leq 1 + 1 \leq 2$, so the maximum is attained at this point. There's no need to search in the interior because of MMT, so you are done.

Indeed, this is a nice application of Maximum MT , because you can restrict finding maximums of non-constant functions to the boundary of a given domain , rather than search the whole domain. If the boundary is nice, then this makes the search much easier.