Complex inequality having $3$ variables

Question

If $x,y,z>0.$ Then prove that $\displaystyle \frac{yz}{x}+\frac{zx}{y}+\frac{xy}{z}\geq x+y+z$

what i try

put $\displaystyle x=a/b,y=b/c,z=c/a$ and $xyz=1$

Then $\displaystyle \frac{b^2}{a^2}+\frac{c^2}{b^2}+\frac{a^2}{c^2}\geq \frac{(a+b+c)^2}{a^2+b^2+c^2}$ (Titu,s lima)

How do i solve it Help me please

Also: https://math.stackexchange.com/q/576592/42969. – Martin R Mar 27 '19 at 18:37 — Martin R, Mar 27 '19 at 18:37

score 2 · Accepted Answer · answered Mar 27 '19 at 13:19

2

Hint : $ yz/x + xz/y \ge 2z$

answered Mar 27 '19 at 13:19

DeepSea

score 1 · Answer 2 · answered Mar 27 '19 at 13:34

1

It's $$\sum_{cyc}(x^2y^2-x^2yz)\geq0$$ or $$\sum_{cyc}z^2(x-y)^2\geq0.$$

answered Mar 27 '19 at 13:34

Michael Rozenberg

2 Answers2