Complex analysis - maximum/modulus principle

[sweetvirgogirl]

Suppose that f is analytic on a domain D, which contains a simple closed curve lambda and the inside of lambda. If |f| is constant on lambda, then either f is constant or f has a zero inside lambda ...
i am supposed to use maximum/modulus principle to prove it ...

here is my take:
if f is constant, i don't see a reason why |f| wouldn't be constant :)

if f is not constant, then the max/min principle applies ...
meaning |f| can not have any local max/min on D
now i am lost at this point ...

i would also like a little clarification on what f is constant on lambda means, because the way i see it .. .if lambda is a closed loop (say a circle), then how can f be always increasing /decreasing ...? maybe i am misinterpreting the problem

 

[shmoe]

if your curve lambda is a(t), t from 0 to 1 then |f| constant on lambda means |f(a(t))|=K for some constant K and all t. Normally, |f(a(t))| can wobble all over the place as it winds around the loop, it can't be increasing on all of [0,1].

if f is a constant on some set, then any function of f is a constant, specifically it's modulus.

What happens if you have no zero? Consider 1/f.

 

[sweetvirgogirl]

if your curve lambda is a(t), t from 0 to 1 then |f| constant on lambda means |f(a(t))|=K for some constant K and all t. Normally, |f(a(t))| can wobble all over the place as it winds around the loop, it can't be increasing on all of [0,1].

if f is a constant on some set, then any function of f is a constant, specifically it's modulus.

What happens if you have no zero? Consider 1/f.

thats what i thought ...

but i still have no insight about f having any zero inside lambda ...

like i kinda see it visually ... like i know it makes sense ... but i got no clue how to "prove" it

 

[shmoe]

If there's no zero of f, you can apply maximum modulus principle to both f and 1/f. What happens?

 

[sweetvirgogirl]

If there's no zero of f, you can apply maximum modulus principle to both f and 1/f. What happens?

if i apply max/min modulus to both f and 1/f, then it means both f and 1/f don't have a local max/min ... so what does it tell me?
i'm sorry ... i don't know if it's lack of confidence or what .. but i still don't see it

 

[shmoe]

If you have minimum modulus principle as well, then you can forget about 1/f.

Apply min/max modulus to f then. What does this tell you about |f| on the interior compared to it's values on the boudnary? (remember |f| is a continuous function, lambda+interior is a closed and boundeed set)