Let $f(z)$ be analytic and nonzero in a region R. Show that $|f(z)|$ has a minimum value in R that occurs on the boundary.
I think you should use the Maximum-Modulus Theorem for the function $1/f(z)$
The Maximum-Modulus Theorem
Let $f(z)$ be analytic and nonzero in a region R. Show that $|f(z)|$ has a minimum value in R that occurs on the boundary.
I think you should use the Maximum-Modulus Theorem for the function $1/f(z)$
The Maximum-Modulus Theorem
Since it is not clear what is meant by a "region" in the problem (and careless formulation leads to easy counterexamples), I will rephrase.
Claim. Suppose $f$ is holomorphic and nonzero in open connected bounded set $U$ and is continuous on $\overline{U}$. Then there exists $\zeta\in\partial U$ such that $|f(\zeta)|=\min_{\overline U}|f| \tag1$
Proof The existence of $\zeta \in \overline{U}$ with the property (1) follows by compactness of $\overline{U}$. Suppose $\zeta\in U$. Then $f(\zeta)\ne 0$ by assumption. Since the right-hand side of (1) is nonzero, the function $g(z)=1/f(z)$ is holomorphic in $U$. In terms of $g$ (1) becomes $|g(\zeta)|=\max_{\overline U}|g| \tag2$ which, by the strong maximum principle, implies $g$ is constant. Hence $f$ is constant, and for a constant function the claim is obviously true. $\Box$