Minimum Modulus Principle for a constant fuction in a simple closed curve Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is constant or $f$ has a zero inside $\gamma$
Here is my take:
if $f$ is constant, i dont see a reason why $|f|$ wouldnt be constant :)
if $f$ is not constant, then the max/min modulus principle applies ...
meaning $|f|$ can not have any local max/min on D
now i am lost at this point ...
 A: Not sure what you mean by the min/max principle. I have only heard of the maximum modulus principle, here is a simple version "given a function analytic on a region containing compact $K$ then the maximum of the modulus is attained on the boundary of $K$". 
How about the minimum, can the minimum modulus be attained not on the boundary? Certainly. This is easily seen by functions that have zeros inside of $K$. Such functions have achieved the smallest possible modulus, namely zero. 
So we extend the question, if a function is not zero in $K$, then can the minimum modulus be attained not on the boundary? The answer is No. This follows because if the function is never zero in $K$ then the reciprocal of the function is also analytic in $K$ and thus enjoys the maximum modulus principle. I guess, we would call this a minimum principle, but really it is a maximum principle for reciprocals. More importantly, it doesn't apply unless the function has no zeros. 
Hopefully, this addresses your concerns, although I am unsure if you asked a question. 
