Complex analysis - maximum/modulus principle

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Discussion Overview

The discussion revolves around the maximum/modulus principle in complex analysis, specifically addressing the conditions under which a function \( f \) that is analytic on a domain \( D \) and constant on a closed curve \( \lambda \) implies that \( f \) is either constant throughout \( D \) or has a zero inside \( \lambda \). Participants explore the implications of these conditions and seek clarification on the principles involved.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that if \( f \) is constant, then \( |f| \) must also be constant, but expresses confusion about the implications if \( f \) is not constant.
  • Another participant clarifies that \( |f| \) being constant on \( \lambda \) means that for a parameterization \( a(t) \), \( |f(a(t))| = K \) for some constant \( K \), and notes that \( |f| \) cannot be monotonically increasing or decreasing over the loop.
  • There is a discussion about the implications of \( f \) having no zeros, with a suggestion to consider \( 1/f \) and its relationship to the maximum modulus principle.
  • Participants express uncertainty about how to prove the existence of a zero inside \( \lambda \) and seek guidance on applying the maximum modulus principle to both \( f \) and \( 1/f \).
  • One participant mentions the minimum modulus principle and suggests focusing on \( f \) alone to compare the values of \( |f| \) on the boundary and the interior of \( \lambda \).

Areas of Agreement / Disagreement

Participants exhibit uncertainty and confusion regarding the application of the maximum/modulus principle and the implications of having no zeros in \( f \). There is no consensus on how to prove the existence of a zero inside \( \lambda \ or on the interpretation of the principles involved.

Contextual Notes

Participants express limitations in their understanding of the maximum/modulus principle and its application, particularly in relation to the conditions of the problem and the implications of having a constant modulus on a closed curve.

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

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