3

Show that $$\frac xy + \frac yz + \frac zx \ge 1 + \frac {z + x}{x + y} + \frac {x + y}{z + x}$$ for $x, y, z \gt 0$.

I observed that this is a homogeneous inequality so normalization might work. I tried to set $x = 1$ or $xyz = 1$ or $x + y + z = 1$, but none of these yields a solution. How do I tackle this problem? In particular, what can I do with the term $1$? Any hints will be appreciated.

Yuxiao Xie
  • 8,536

1 Answers1

4

Using CS inequality in slightly different way, $$\frac{x}y + \frac{y}z + \frac{z}x + 1 \geqslant \frac{(x+y+z+x)^2}{xy+yz+zx+x^2} = \frac{(\color{red}{x+y}\:+\:\color{blue}{z+x})^2}{(\color{red}{x+y})\cdot(\color{blue}{z+x})} = \frac{x+y}{z+x}+2 + \frac{z+x}{x+y}$$

The cyclic counterparts of the inequality can be shown similarly.

Addendum: the inequality above is perhaps seen simplest by $$\frac{x}y + \frac{y}z + \frac{z}x + 1 = \frac{x^2}{xy}+\frac{y^2}{yz}+\frac{z^2}{zx}+\frac{x^2}{x^2} \geqslant \frac{(x+y+z+x)^2}{xy+yz+zx+x^2} $$ the CS form is called "Titu's lemma" and is a useful tool to have.

Macavity
  • 46,381
  • Very witty ! But, two things: 1) for some (young and less young) readers, not accustomed to the tricks of the trade, I add that you take Cauchy-Schwarz inequality with vectors $(\sqrt{\dfrac{x}{y}},\sqrt{\dfrac{y}{z}},\sqrt{\dfrac{z}{x}},1)$ and $(\sqrt{xy},\sqrt{yz},\sqrt{zx},x)$ of $\mathbb{R}^4$ 2) the case of equality can be verified to hold iff $x=y=z$. – Jean Marie Apr 25 '16 at 12:21
  • @JeanMarie Personally I believe in giving just enough information for those interested to finish solving the problem. Catering to all kinds of readers is a huge and largely unnecessary task, but normally if anyone has a follow up query I do answer. – Macavity Apr 25 '16 at 12:26
  • I understand very well this point of view. In this case, my opinion is that we are at the limit of understandability for - say - an undergraduate full of good will. – Jean Marie Apr 25 '16 at 12:40
  • @JeanMarie Added a bit more, hopefully this will be enough explanation. – Macavity Apr 25 '16 at 13:06