Prove the inequality.

Question

$$\left(\sum_{i=1}^na_i^p\right)^{1/p} \ge \left(\sum_{i=1}^na_i^q\right)^{1/q} $$ if $0 < p \le q$ for $a_i\ge 0$. I have proved that the inequality holds for $ p=q $ (trivial) and i have also proved that it holds if one of the two sums is equal to 1, but I don't know how to continue. Please help.

Answer 1

Hint: use the Jensen inequality with $x\to x^{q/p}$.

Answer 2

Set $b_i=a_i^q$ and $t=p/q\leq1$, then enough to show $$ \sum_{i=1}^nb_i^t\geq (\sum_{i=1}^nb_i)^t. $$ i.e. $$ \sum_{i=1}^n\left(\frac{b_i}{\sum_{i=1}^nb_i}\right)^t\geq1. $$ Note that $$ \sum_{i=1}^n\left(\frac{b_i}{\sum_{i=1}^nb_i}\right)^t\geq\sum_{i=1}^n\left(\frac{b_i}{\sum_{i=1}^nb_i}\right)=1. $$ The first inequality is becuase $x^t\geq x$ for $t\leq 1$ and $x\leq 1$.