Inegalitate nice

[Claudiu Mindrila]

Daca \(
a,b,c \in (0; + \infty )
\), astfel incat \(
a^6 + b^6 + c^6 = 1

\), sa se arate ca: \(
\left( {a + b + c + d} \right)^2 \leq \frac{{b^2 c^2 d^2 }}
{{a^{10} }} + \frac{{a^2 c^2 d^2 }}
{{b^{10} }} + \frac{{a^2 b^2 d^2 }}
{{c^{10} }} + \frac{{a^2 b^2 c^2 }}
{{d^{10} }}

\).

Alexandru Negrescu, Axioma, nr. 23

[Marius Mainea]

\( LHS=\sum {\frac{(\frac{bcd}{a^2})^2}{a^6}\ge \frac{(\sum {\frac{bcd}{a^2})^2}}{\sum {a^6}} \) si e suficient sa aratam ca

\( \sum {\frac{bcd}{a^2}}\ge \sum {a} \) sau

\( \sum {\frac{1}{a^3}}\ge \sum {\frac{1}{bcd}} \) care este adevarata din inegalitatea ,,rearanjamentelor''.