Proving an Inequality Using Cauchy Formula: Tips and Tricks

[pixel01]

Can anyone help me to prove this inequality:

(a^2)/b+(b^2)/c+(c^2)/a >= a+b+c.

I know i must use Cauchy formula, but can not prove it.

Thank you .

 

[malawi_glenn]

What do you mean, that you don't know how to prove cauchy's inequality? Or that you can not come to that point where you have the equation on the form of cauchy's inequality?

 

[pixel01]

Thank you. I've solved it myself. This is one of the homework series that i have to use cauchy's inequality.

Can you help me to prove this one:

a/(a+b)+b/(b+c)+c/(c+a) > sqrt(a/(b+c)) + sqrt(b/(a+c)) + sqrt(c/(a+b)).

 

[malawi_glenn]

Thank you. I've solved it myself. This is one of the homework series that i have to use cauchy's inequality.

Can you help me to prove this one:

a/(a+b)+b/(b+c)+c/(c+a) > sqrt(a/(b+c)) + sqrt(b/(a+c)) + sqrt(c/(a+b)).

what have you done so far?..

 

[pixel01]

My idea by now is :
a/(a+b)<1, b/(b+c)< 1 and c/(c+a)<1, so the left hand side is smaller than sqrt(a/(a+b))+sqrt(b/(b+c))+sqrt(c/(c+a)).
Then I try to compare the right hand side with sqrt(a/(a+b))+sqrt(b/(b+c))+sqrt(c/(c+a)) because they are both in the square root type (i hope it will be easier). But it doesn't work so far. Can you give me some hints.