(Complex number) I have no idea on this

(Complex number) I have no idea on this :(

Homework Statement

If z lies on circle |z|=2, then show that
$$\left\lvert \frac{1}{z^4-4z^2+3} \right\rvert ≤ \frac{1}{3}$$

The Attempt at a Solution

Please somebody give me an idea. [URL]http://209.85.48.12/11451/115/emo/dizz.gif[/URL]

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Fredrik
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Gold Member

Consider any fraction x/y where x,y>0. If you replace y with something smaller, you make the fraction larger. In other words, if y>u, then x/y<x/u. Can you find something that's smaller than $|z^4-4z^2+3|$?

I haven't verified that that this gives us something useful, but it seems like the obvious place to start.

Can you find something that's smaller than $|z^4-4z^2+3|$?
No. How will i do that?

verty
Homework Helper

Sorry, I didn't notice this was for complex numbers.

If |z| <= 2, |z|^2 = |z^2| <= 4, ...

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Fredrik
Staff Emeritus
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No. How will i do that?
The triangle inequality. $$|z|-|w|\leq|z+w|\leq|z|+|w|$$

The triangle inequality. $$|z|-|w|\leq|z+w|\leq|z|+|w|$$
Okay, but how i will apply it here?

Should it be like this:-
$$|z^4-4z^2+3|≤ |z^4|+|3-4z^2|$$

Fredrik
Staff Emeritus
Gold Member

Okay, but how i will apply it here?

Should it be like this:-
$$|z^4-4z^2+3|≤ |z^4|+|3-4z^2|$$
That inequality is correct, but you need to find something smaller than (or equal to) $|z^4-4z^2+3|$. I don't think I can tell you much more without solving the whole problem for you. I'll just add that I have verified that this approach gives us the correct answer.

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Fredrik
Staff Emeritus
Gold Member

I think it's easier to just use the triangle inequality twice.

That inequality is correct, but you need to find something smaller than (or equal to) $|z^4-4z^2+3|$. I don't think I can tell you much more without solving the whole problem for you. I'll just add that I had verified that this approach gives us the correct answer.
$$|z^4-4z^2+3|≥ |z^4|-|3-4z^2|$$

Fredrik
Staff Emeritus
Gold Member

$$|z^4-4z^2+3|≥ |z^4|-|3-4z^2|$$
Yes. Now do something similar to the second term on the right, to get something even smaller than (or equal to) $|z^4|-|3-4z^2|$.

Yes. Now do something similar to the second term on the right, to get something even smaller than (or equal to) $|z^4|-|3-4z^2|$.
How will i do that? How will i apply the triangle inequality?

Note that z4-4z2+3=(z2-1)(z2-3), and use de Moivre's formula.

ehild
I tried that but i am not able to understand how would i apply De Moivre's theorem?

What is te magnitude of z2?

ehild
Magnitude of z2? Now you are making me confused (which i am already).

Magnitude is the same as absolute value. What is it for z?

ehild
For z its |z| or $\sqrt{x^2+y^2}$
and for z2, it is $x^2+y^2$

Right..?

I like Serena
Homework Helper

Hi Pranav-Arora!

Can you also write z as an exponential power?

So what can you say about |z2|?

Hi Pranav-Arora!

Can you also write z as an exponential power?

So what can you say about |z2|?
Hi ILS! :)

|z|=2
therefore |z2|=4. :)

I like Serena
Homework Helper

Yep! [URL]http://i154.photobucket.com/albums/s271/R2Garnets/Smileys/Smiley-Duh.gif[/URL]

So what can you say about |z2-1|?

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Fredrik
Staff Emeritus
Gold Member

How will i do that? How will i apply the triangle inequality?
If I tell you that, I will have completely solved the problem for you. Isn't the 99% I've already done enough? You have to try something yourself.

Have you tried anything at all? I only gave you two inequalities, so if the first one you tried didn't work, it has to be the other one. It looks like you're not even trying, and just want someone else to do 100% of the work for you.

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Yep! [URL]http://i154.photobucket.com/albums/s271/R2Garnets/Smileys/Smiley-Duh.gif[/URL]

So what can you say about |z2-1|?
Should it be equal to 3?

If I tell you that, I will have completely solved the problem for you. Isn't the 99% I've already done enough? You have to try something yourself.

Have you tried anything at all? I only gave you two inequalities, so if the first one you tried didn't work, it has to be the other one. It looks like you're not even trying, and just want someone else to do 100% of the work for you.
No Fredrik, i am trying myself too. But i am really not able to work out what should i do next?
Complex number is the weakest point of mine in mathematics and it is really really hard for me. I try to do my best but if you think that you have told me the 99% of the solution, i will try it again. :)

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I like Serena
Homework Helper

Should it be equal to 3?
No. How did you get to 3?

No. How did you get to 3?
I thought |z2|=4.
or |z2|-1=3

Since |z2|-1=|z2-1|
therefore |z2-1|=3

EDIT: I have realised that |z2|-1=|z2-1| does not hold true at 0.

What should i do now? :(

I like Serena
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Since |z2|-1=|z2-1|
This is not true.
Why isn't it?

I like Serena
Homework Helper

Consider that z is not a real number, but a complex number.
It is represented by a 2D vector.
The absolute value is not the regular absolute value of real numbers, but the length of the vector that z represents.

This is not true.
Why isn't it?
Because it does not hold true at 0.

Fredrik
Staff Emeritus