# (Complex number) I have no idea on this

1. Oct 29, 2011

### Saitama

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

1. The problem statement, all variables and given/known data
If z lies on circle |z|=2, then show that
$$\left\lvert \frac{1}{z^4-4z^2+3} \right\rvert ≤ \frac{1}{3}$$

2. Relevant equations

3. The attempt at a solution

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

Last edited by a moderator: Apr 26, 2017
2. Oct 29, 2011

### Fredrik

Staff Emeritus
Re: (Complex number) I have no idea on this :(

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.

3. Oct 29, 2011

### Saitama

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

No. How will i do that?

4. Oct 29, 2011

### verty

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

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

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

Last edited: Oct 29, 2011
5. Oct 29, 2011

### Fredrik

Staff Emeritus
Re: (Complex number) I have no idea on this :(

The triangle inequality. $$|z|-|w|\leq|z+w|\leq|z|+|w|$$

6. Oct 29, 2011

### Saitama

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

Okay, but how i will apply it here?

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

7. Oct 29, 2011

### Fredrik

Staff Emeritus
Re: (Complex number) I have no idea on this :(

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.

Last edited: Oct 29, 2011
8. Oct 29, 2011

### Fredrik

Staff Emeritus
Re: (Complex number) I have no idea on this :(

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

9. Oct 29, 2011

### Saitama

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

$$|z^4-4z^2+3|≥ |z^4|-|3-4z^2|$$

10. Oct 29, 2011

### Fredrik

Staff Emeritus
Re: (Complex number) I have no idea on this :(

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

11. Oct 29, 2011

### Saitama

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

How will i do that? How will i apply the triangle inequality?

12. Oct 29, 2011

### Saitama

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

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

13. Oct 29, 2011

### Saitama

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

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

14. Oct 29, 2011

### Saitama

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

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

Right..?

15. Oct 29, 2011

### I like Serena

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

Hi Pranav-Arora!

Can you also write z as an exponential power?

So what can you say about |z2|?

16. Oct 29, 2011

### Saitama

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

Hi ILS! :)

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

17. Oct 29, 2011

### I like Serena

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

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

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

Last edited by a moderator: Apr 26, 2017
18. Oct 29, 2011

### Fredrik

Staff Emeritus
Re: (Complex number) I have no idea on this :(

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.

Last edited: Oct 29, 2011
19. Oct 29, 2011

### Saitama

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

Should it be equal to 3?

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. :)

Last edited by a moderator: Apr 26, 2017
20. Oct 29, 2011

### I like Serena

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

No. How did you get to 3?