พิจารณา$$\frac {a^2}{b}+\frac {b^2}{c}+\frac {c^2}{a} +(a+b+c) \geq 6(\frac {a^2+b^2+c^2}{a+b+c})$$
$\Longleftrightarrow \frac {a^2}{b}-2a+b+\frac {b^2}{c}-2b+c+\frac {c^2}{a}-2c+a \geq 6(\frac {a^2+b^2+c^2}{a+b+c})-2(a+b+c) $
$\Longleftrightarrow \frac {(a-b)^2}{b}+\frac {(b-c)^2}{c}+\frac {(c-a)^2}{a} \geq 2(\frac {(a-b)^2+(b-c)^2+(c-a)^2}{a+b+c}) $
้homogenize ให้ $a=x+y , b=y+z , c=z+x$
$\Longleftrightarrow \frac {(x-z)^2}{x+y}+\frac {(y-x)^2}{y+z}+\frac {(z-y)^2}{z+x} \geq (\frac {(x-z)^2+(y-x)^2+(z-y)^2}{x+y+z}) $
ซึ่งพิจารณา $\frac {(x-z)^2}{x+y}+\frac {(y-x)^2}{y+z}+\frac {(z-y)^2}{z+x} \geq \frac {(x-z)^2}{x+y+z}+\frac {(y-x)^2}{x+y+z}+\frac {(z-y)^2}{x+y+z}$
ซึ่งได้ตามต้องการ
