Keenan Santos

2022-07-16

Is it true for $2\le q\le p$ that
$‖x{‖}_{q}\le {n}^{\frac{p-q}{pq}}‖x{‖}_{p}$
where x is an n-dimensional vector. I only need the inequality for n=2, so that would suffice. I'm just curious if it's true for any n. Just from plugging it into a graphing calculator I believe it to be true, but how would I prove it?

Sophia Mcdowell

Expert

$‖x{‖}_{q}^{p}={\left(\frac{1}{n}\sum _{i=1}^{n}|{x}_{i}{|}^{q}\right)}^{p/q}{n}^{p/q}$
and using convexity of the map $t↦{t}^{p/q}$, one gets
$‖x{‖}_{q}^{p}⩽\frac{1}{n}\sum _{i=1}^{n}|{x}_{i}{|}^{p}{n}^{p/q}$
or in other words,
$‖x{‖}_{q}^{p}⩽{n}^{p/q-1}‖x{‖}_{p}^{p}.$

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