# need help to prove inequalities

• Sep 2nd 2011, 04:07 AM
Mhmh96
need help to prove inequalities
Hello..

I need help to prove this inequalities (Worried)
http://a7.sphotos.ak.fbcdn.net/hphot..._6695647_n.jpg
• Sep 4th 2011, 01:50 PM
Aryth
Re: need help to prove inequalities
I don't know how good you are at inequalities but it basically goes like this:

$\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} =$ $\frac{a^2cd}{abcd} + \frac{b^2ad}{abcd} + \frac{c^2ab}{abcd} + \frac{d^2bc}{abcd}$

$= \frac{a^2cd + b^2ad + c^2ab + d^2bc}{abcd} \leq \frac{a^2 + b^2 + c^2 + d^2}{abcd}$ $\leq \frac{a + b + c + d}{abcd} \leq \frac{4}{abcd}$

So, $\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \leq \frac{4}{abcd}$
• Sep 4th 2011, 01:55 PM
alexmahone
Re: need help to prove inequalities
Quote:

Originally Posted by Aryth
$= \frac{a^2cd + b^2ad + c^2ab + d^2bc}{abcd} \leq \frac{a^2 + b^2 + c^2 + d^2}{abcd}$ $\leq \frac{a + b + c + d}{abcd}$

Care to explain the first inequality?

The second inequality is false. Take a = 3, b = 0.3, c = 0.3, d = 0.4 for a counterexample.