Question

Aratati ca: a la puterea 4/b + b la puterea 4/c + c la puterea 4/a > ab^+bc^+ca^,pentru a,b,c>0.

Answer 1

Folosim inegalitatea dintre media artimetica si media geometrica.
$M_a \leq M_g<=> \frac{a+b}{2} \leq \sqrt{a\cdot b} <=>a+b \leq 2 \sqrt{a\cdot b}$
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]