Inegalitate cu o altfel de conditie

[Radu Titiu]

Fie \( a,b,c >0 \) a.i. \( (a-b)^2+(b-c)^2+(c-a)^2\leq 2(1-\sqrt[3]{a^2b^2c^2}) \). Demonstrati ca:

\( a^3+b^3+c^3 \leq a+b+c \)

[Marius Mainea]

Condititia este echivalenta cu

\( \sum {a^2}+3\sqrt[3]{a^2b^2c^2}\le \sum {ab}+1 \)
iar concluzia cu
\( (a+b+c)(\sum {a^2}-\sum {ab}-1)+3abc\le 0 \)
Asadar
\( LHS\le (a+b+c)(-\sqrt[3]{a^2b^2c^2})+3abc\le 0 \)
ultima fiind echivalenta cu
\( (a+b+c)^3\ge 27 abc \)