Question

8. Let f be analytic on a bounded domain D and continuous on Du B, where B is the boundary of D. Show that if f is never zero on D, then the minimum of lfl is assumed on B. You will need to use the fact that lfl does, indeed, assume a minimum somewhere on D B

Answer 1

Solution:

Since f(z) is analytic on D, with no zeros on D, then the function $\frac{1}{f\left ( z \right )}$ is also analytic on D.

Thus, If |f(z)| attains its minimum on D, then $\left | \frac{1}{f\left ( z \right )} \right |$ attains its maximum on D

This implies that $\left | \frac{1}{f\left ( z \right )} \right |$     is constant by the Complex Version of the Strict Maximum Principle.

Now, similarly, we look at the function $\left | \frac{1}{f\left ( z \right )} \right |$    this is continuous on D ∪ ∂D, so it attains a maximum somewhere. The maximum can’t be in D unless f(z) is constant, thus we conclude that , |f(z)| attains its minimum on ∂D.

Note:- here B=∂D.