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Let $D$ be a bounded region and $f$ is an analytic function on $D$. Show that if there is a constant $c ≥ 0$ such that $|f(z)| = c$ for all $z$ in the boundary of $D$ then either $f$ is a constant function or $f$ has a zero in $D$.

Is this just a direct consequence of the Maximum-Modulus Theorem?

user274933
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  • That depends on your definition of "direct". For a sufficiently narrow definition, it's an indirect consequence. – Daniel Fischer Oct 16 '15 at 12:44
  • Can you elaborate on that please? If it's not, then how would I go about showing it? – user274933 Oct 16 '15 at 12:47
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    It is a consequence of the maximum modulus theorem. The question is whether you call it direct or not. To show it, make a case distinction between $c = 0$ and $c > 0$. For $c = 0$, it is a direct consequence of the maximum modulus theorem that $f \equiv 0$. For $c > 0$, suppose $f$ has no zero in $D$. Then apply the maximum modulus theorem to something in order to deduce that $f$ is constant. – Daniel Fischer Oct 16 '15 at 12:51
  • $f$ is only defined in $D$, so what does $f$ on the boundary mean? – zhw. Oct 16 '15 at 16:15
  • @zhw. $f$ is defined on the boundary of D to be a constant function s.t. $f$ is continuous on the closure of D. The last part was forgotten to be mentioned by the author. – Emo Jan 04 '19 at 13:30

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