variant of young's inequality

Question

Let $\alpha_{i}\geq 0$ y $p_{i}\geq 0$ for $i=1,2,...,n$ such that: $$\frac{1}{p_1}+\frac{1}{p_2}+...+\frac{1}{p_n}=1$$ proof that:

$$\alpha_1\alpha_2...\alpha_n\leq \frac{\alpha_1^{p_1}}{p_1}+\frac{\alpha_2^{p_2}}{p_2}+...+\frac{\alpha_n^{p_n}}{p_n}$$ I have been trying to perform this test analogously to the proof of Young's inequality but I have not been able to get it

Answer 1

Hint: You can prove this via induction over $n$.