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Suppose z is holomorphic in a bounded domain, continuous on the closure, and constant on the...

Suppose z is holomorphic in a bounded domain, continuous on the closure, and constant on the boundary. Prove that f must be constant throughout the domain.

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Answer #1

Given ,  z is holomorphic in a bounded domain, continuous on the closure, and constant on the boundary .

By Maximum moduler principal moduler principal every holomorphic function attains its maximum and minimum on its boundary .

Now since the function is constant on its boundary so suppose ,

f (z) = c on its boundary where c is a constant .

by maximum moduler principal maximum of the function is c and also minimum is c

for all z belongs to the domain .

f(z) = c for all z belongs to the domain .

Hence  f must be constant throughout the domain .

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If you have any doubt please comment .

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