# Thread: need help to prove inequalities

1. ## need help to prove inequalities

Hello..

I need help to prove this inequalities

2. ## Re: need help to prove inequalities

I don't know how good you are at inequalities but it basically goes like this:

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

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

So, $\displaystyle \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \leq \frac{4}{abcd}$

3. ## Re: need help to prove inequalities

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