Folosim inegalitatea dintre media artimetica si media geometrica.
[tex]M_a \leq M_g<=> \frac{a+b}{2} \leq \sqrt{a\cdot b} <=>a+b \leq 2 \sqrt{a\cdot b} [/tex]
Folosind ultima inegalitate obtinem:
[tex] \frac{a^4}{b} +bc^2 \leq 2a^2c\\
\frac{b^4}{c} +ca^2 \leq 2b^2a\\
\frac{c^4}{a} +ab^2 \leq 2c^2b\\[/tex]
Insumam inegalitatile:
[tex] \frac{a^4}{b} +bc^2+ \frac{b^4}{c} +ca^2+ \frac{c^4}{a} +ab^2 \leq 2a^2c+2b^2a+2c^2b\\
\frac{a^4}{b} + \frac{b^4}{c} + \frac{c^4}{a} \leq a^2c+b^2a+c^2b[/tex]
