Prove that $||x||_{p} \leq ||x||_{q} \cdot d^{(1/p) - (1/q)}$, if $1 < p < q $ for any $x \in \mathbb R^{d}$

Question

Prove that $||x||_{p} \leq ||x||_{q} \cdot d^{(1/p) - (1/q)}$, if $1 < p < q $ for any $x \in \mathbb R^{d}$

How do you prove this using Holder's inequality?

Answer 1

For $\mathbf{x} \in \mathbb{R}^d$ and $1 < p < q < \infty$ you get:

$$\|\mathbf{x} \|_p^p = \sum_{n=1}^d |x_n|^p \stackrel{\text{Höl}}{\leq} \| \mathbf{1}\|_{(q/p)^\prime}\ \|\mathbf{x}\|_q^p = d^{1 - p/q}\ \|\mathbf{x}\|_q^p$$

where $\mathbf{1} =(1, \ldots ,1)$ and $(q/p)^\prime$ is the Hölder conjugate of $q/p >1$, therefore:

$$\|\mathbf{x} \|_p \leq d^{1/p - 1/q}\ \|\mathbf{x}\|_q \;,$$

as you wanted.