Inequality problem involving QM-AM-GM-HM or Cauchy Schwarz inequality Question: Prove that if $x$, $y$, $z$ are positive real numbers then the following inequality holds: $$\frac{x+y}{x^2+y^2}+\frac{y+z}{y^2+z^2}+\frac{z+x}{z^2+x^2}\leq\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$I tried thinking of applying QM-AM, but I didn't know what variables to use. I also thought that this could be related to Cauchy's inequality, but I wasn't sure, again, of what variables to use. Any help would be greatly appreciated, thanks!

We want to show that:
$$\frac{x+y}{x^2+y^2}+\frac{y+z}{y^2+z^2}+\frac{z+x}{z^2+x^2}\leq\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$
Note that, by the QM-AM Inequality:
$${\left(\frac{x+y}{2}\right)}^2 \leq \frac{x^2 + y^2}{2}.$$
Consequently:
$$\frac{x + y}{x^2 + y^2} \leq \frac{2}{x + y}.$$
Therefore, it remains to show that:
$$\frac{2}{x + y} + \frac{2}{y + z} + \frac{2}{x + z} \leq \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.$$
By the AM-HM Inequality:
$$\frac{2}{\frac{1}{x} + \frac{1}{y}} \leq \frac{x + y}{2}.$$
Therefore:
$$\frac{2}{x + y} \leq \frac{\frac{1}{x} + \frac{1}{y}}{2}$$
Similarly:
$$\frac{2}{y + z} \leq \frac{\frac{1}{y} + \frac{1}{z}}{2},$$
and
$$\frac{2}{x + z} \leq \frac{\frac{1}{x} + \frac{1}{z}}{2}.$$
QED.
