An analytic function f such that |f(a)| is less than or equal to |f(z)| for all z in G.Then f is .

Question

Suppose that f:G to C is analytic , where G is a region and a belongs to G. Also |f(a)|<_|f(z)| for all z in G. Show that f(a) =0 or f is constant.

Answer 1

If $f(a)\ne 0$, then $f(z)\ne 0$ for all $z\in G$, hence $1/f$ is analytic. Apply the maximum principle.

Answer 2

If $f(a)\not=0$, we must have $f$ bounded away from $\{0\}$. Then $\frac1f$ can be extended to a bounded entire function. Hence constant, by Liouville's theorem.

Note: The function $\frac1f$ can be extended because the zeros of an analytic function are isolated; hence all the singularities are removable.