Is the Cauchy Schwarz inequality generalizable to any natural power $ p \in \mathbb{N}$?

Asked Oct 18 '21 at 14:28

Active Oct 18 '21 at 14:40

Viewed 155 times

I was wondering if we let $ p $ be a natural number, would the following hold ?

$$ \left( \sum_{k=1}^{n}x_ky_k \right)^p \leq \sum_{k=1}^{n}x_k^p\sum_{k=1}^{n}y_k^p $$

I believe it should. If we reformulate this in terms of dot product, we would have:

$$ \Vert \vec{x}\cdot \vec{y} \Vert^p \leq \Vert \vec{x} \Vert^p \cdot \Vert \vec{y} \Vert^p $$

which 'seems' true.

Please notify if anything isn't clear or doesn't comply with the rules.

Thanks

edited Oct 18 '21 at 14:40

J. W. Tanner

asked Oct 18 '21 at 14:28

sleepydrmike

2

Check this: https://math.stackexchange.com/questions/484872/why-doesnt-cauchy-schwarz-in-mathbbrn-generalize-to-exponents-k2. – Martin R Oct 18 '21 at 14:35
How does it “seem” true or why do you “believe it should”? – Martin R Oct 18 '21 at 14:41
@MartinR Thank you for the reference. Wrong intuition I guess – sleepydrmike Oct 18 '21 at 14:42
It is true for $0 < p \le 2$ though ... – Martin R Oct 18 '21 at 14:43

0 Answers0