Let f(z) and g(z) be analytic functions defined on a bounded domain D and continuous on D and its boundary ∂D. Suppose that g(z)

=0∀z∈D∪∂D. Prove that if the inequality ∣f(z)∣≤∣g(z)∣ holds on all z∈∂D, then it also holds for all z∈D.