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$\dfrac{1}{2}(\dfrac{a^2+b^2}{c}+\dfrac{b^2+c^2}{a}+\dfrac{c^2+a^2}{b})$

\geq $\dfrac{ab}{c}+\dfrac{bc}{a}+\dfrac{ca}{b}$ \geq a+b+c (Cauchy)

Cũng theo Cauchy:

$a^4+a^4+a^4+b^4$ \geq$4\sqrt[4]{a^{12}b^4}$=$4a^3b$

$a^4+b^4+b^4+b^4$ \geq$4ab^3$ ; $b^4+b^4+b^4+c^4$\geq$4b^3c$

$c^4+c^4+c^4+b^4$ \geq$4bc^3$ ; $c^4+c^4+c^4+a^4$\geq$4c^3a$

$c^4+a^4+a^4+a^4$ \geq$4ca^3$

\Rightarrow $8(a^4+b^4+c^4)$\geq$4(a^3b+ab^3+b^3c+bc^3+c^3a+ca^3)$

\Leftrightarrow $a^4+b^4+c^4$\geq$\dfrac{1}{2}(a^3b+ab^3+b^3c+bc^3+c^3a+ca^3)$

\Leftrightarrow $\dfrac{a^4+b^4+c^4}{abc}$\geq $\dfrac{1}{2}$ ($\dfrac{ab(a^2+b^2)+bc(b^2+c^2)+ca(c^2+a^2)}{abc}$)

\Leftrightarrow $\dfrac{a^3}{bc}+\dfrac{b^3}{ac} + \dfrac{c^3}{ab}$ \geq $\dfrac{1}{2}(\dfrac{a^2+b^2}{c}+\dfrac{b^2+c^2}{a}+\dfrac{c^2+a^2}{b})$

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