Inegalitate usurica

[Claudiu Mindrila]

\( \left|\begin{array}{c}
a_{1},\ a_{2},\dots,\ a_{n}\in\mathbb{R},\ \lambda\in\left(0,\ \infty\right),\ n\in\mathbb{N},\ n\ge2\\
a_{1}^{2}+a_{2}^{2}+\dots+a_{n}^{2}\le\lambda^{2}\end{array}\right|\Longrightarrow a_{1}^{3}+a_{2}^{3}+\dots+a_{n}^{3}\le\lambda^{3}. \)

[Marius Mainea]

Indicatie:

Daca \( x\in[0,1]\Longrightarrow x^3\le x^2 \)

[Claudiu Mindrila]

\( \sum_{i=1}^{n}a_{i}\le\lambda^{2}\Longrightarrow a_{1},\ a_{2},\dots,\ a_{n}\le\lambda\Longrightarrow\sum_{i=1}^{n}a_{i}^{2}\left(\lambda-a_{i}\right)\ge0\Longrightarrow\sum_{i=1}^{n}a_{i}^{3}\le\lambda^{3} \).