Math Is Fun Forum

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Questions

Well, there is a corollary on Maximum Modulus Principle that goes like this:

"If

is analytic on a path connected open set

and

attains its maximum value at a point

in

, then

is constant on

."

Well my question are:

1.)Does this statement holds true if I replace "maximum" with "minimum"?

2.)More generally, does Maximum Modulus Principle and all its corollaries the same as Minimum Modulus Principle just that we only have to replace the word "maximum" with "minimum"?

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3.) Alright this question is not on Maximum Modulus Principle, it's something regarding the phrasing of Mathematics that I am confused in.

Let's say if we say something like

lies in

is it the same as saying

lies on

?

and the same goes for if

lies in

is it the same as saying

lies on

?

I am asking this because I always though if we say something lie in something such as 

we mean that it is contain inside the circle. Whereas when we say something lies on

, we actually mean that it "sit" right on the circumference of the circle or sometime refer to as

.

However, it seem to me that some textbook just use them interchangeable, could any kind soul confirm this for me?

Thanks and will appreciate if anyone could answer my questions as my exam on Complex Analysis is on next week

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Questions

I think if you have f as the identity map and A as the unit circle around the origin, then |f(z)| has its minimum at the origin but isn't constant, so that's a counterexample.
So therefore, your question 2 is wrong as well.

I don't think there are any standard definitions for "in" or "on".
Probably best to use ∈and other recognised notation to say exactly what you mean.

---

Why did the vector cross the road?
It wanted to be normal.

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Re: Questions

Thanks!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Questions

The minimum modulus principle holds, but the theorem must add the case that mathsyperson pointed out:

If f attains it's minimum inside a region G, and f is never zero, then f is constant.

It is a good exercise to prove this.  As for "in" and "on", they are used interchangeably.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

coffeeking

Member

Registered: 2007-11-18

Posts: 44

Re: Questions

I think I get the idea. Thanks