Prove the inequality $\|x\|_p \le N^{1 - 1/q}\|x\|_q$.

Question

In my functional analysis course, we define $$\|x\|_p = \left(\sum_{k = 1}^N |x_i|^p\right)^{1/p}$$ for $x \in \mathbb{R}^N$ and $p \in [1, +\infty)$. Then the teacher told us that, for $p \le q$, it was obvious that $$\|x\|_q \le \|x\|_p \le N^{1 - 1/q}\|x\|_q.$$ The first inequality is not too hard, because $$\left|\frac{x_j}{\|x\|_q}\right|^q \le \left|\frac{x_j}{\|x\|_q}\right|^p \quad \forall j \in \{1, \ldots, N\},$$ and then, after summation $1 \le (\|x\|_p/\|x\|_q)^p$. But I don't really see how can I prove the second inequality. Any idea?

Answer 1

Observe that $$ \|x\|_p^q = N^{q/p}\left(\frac 1N\sum_{k = 1}^N |x_i|^p\right)^{q/p}. $$ Then use Jensen's inequality $\varphi\left(\sum_{k=1}^N\alpha_kt_k\right)\leqslant \sum_{k=1}^N\alpha_k\varphi\left(t_k\right)$ where $\alpha_k\geqslant 0$ and $\sum_{k=1}^N\alpha_k=1$, here with $\alpha_k=1/N$ and $\varphi(t)=t^{q/p}$.