need help to prove inequalities

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September 2nd 2011, 03:07 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
September 4th 2011, 12:50 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, $\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \leq \frac{4}{abcd}$
September 4th 2011, 12:55 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.

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