Inegalitate.

[Claudiu Mindrila]

Sa se demonstreze inegalitatea: \( \frac{x^{3}}{y^{3}}+\frac{y^{3}}{z^{3}}+\frac{z^{3}}{x^{3}}\ge\frac{1}{2}\left(\frac{x^{2}}{yz}+\frac{y^{2}}{zx}+\frac{z^{2}}{xy}\right)+\frac{3}{2},\ \forall x,\ y,\ z\in\left(0,\ +\infty\right) \).

[Marius Mainea]

Folosind inegalitatea lui Schur

\( \sum\frac{x^3}{y^3}+3\ge\sum_{sym}(\frac{x}{y})^2(\frac{y}{z})=\sum_{cyc}\frac{x^2}{yz}+\sum_{cyc}\frac{yz}{x^2}\ge RHS+3 \)