Thread: [SOLVED] Minimum of a complex function, proof

I've no idea what theorem can be useful to prove the following : Let be a continuous function in a closed and bounded region , assume also that is analytic and not constant in the interior of .
Assuming that in all , prove that has a minimum and that it is reached in the frontier of and never in the interior of .

I'm not sure, but I suspect this may be an application of Liouville's theorem:

http://www.proofwiki.org/wiki/Liouvi...mplex_Analysis)

... check it out anyway, there may be techniques and insights in there which will give you some inspiration.

If , then what can you say about the (analytic) function using the Maximum Modulus Principle?

Originally Posted by shawsend

If , then what can you say about the (analytic) function using the Maximum Modulus Principle?

That reaches a minimum in the frontier of .
Not sure I'm right. By the way I never heard this principle before. That's interesting, it reminds me of Extreme value theorem - Wikipedia, the free encyclopedia. Thanks a lot!

By the Maximum Modulus Principle, reaches a maximum on the boundary which means that must reach a minimum on the boundary as well.