show that $\Big(\sum_{i=1}^{n}\alpha_i ^2\Big)^2\leq\Big(\sum_{i=1}^{n}\alpha_i \Big)\Big(\sum_{i=1}^{n}\alpha_i ^3\Big)$

Question

Let $\alpha_1,\alpha_2,...,\alpha_n>0.$ How can I show that $$\Big(\sum_{i=1}^{n}\alpha_i ^2\Big)^2\leq\Big(\sum_{i=1}^{n}\alpha_i \Big)\Big(\sum_{i=1}^{n}\alpha_i ^3\Big).$$

Please provide me a hint to start.

Answer 1

Write $\alpha_i^2=\alpha_i^{\frac{1}{2}}\alpha_i^{\frac{3}{2}}$, and then use the Cauchy-Schwarz inequality.