When does the equality hold in the Holder inequality?

Question

I am considering the series case. In the Holder inequality, we have $$\sum|x_iy_i|\leq\left(\sum|x_i|^p\right)^{\frac1p} \left(\sum|y_i|^q\right)^{\frac1q},$$ where $\frac1p+\frac1q=1,~p, q>1$.

In Cauchy inequality (i.e., $p=q=2$), I know that the equality holds if and only if $x$ and $y$ are linearly dependent. I am wondering when the equality holds in the Holder inequality.

See e.g. the last three lines of the proof as written on Wikipedia. — Clement C., Jul 01 '18 at 20:05
Thanks. I am considering the series instead of the integral here. Would it be different? — coolcat, Jul 01 '18 at 20:08
No, it's basically the same. Series are integrals with the counting measure. — Clement C., Jul 01 '18 at 20:09
When $(|x_1|^p,|x_2|^p,...,|x_n|^p)||(|y_1|^q,|y_2|^q,...,|y_n|^q).$ — Michael Rozenberg, Jul 01 '18 at 20:15

score 5 · Accepted Answer · answered Jul 01 '18 at 20:20

The Hölder inequality comes from the Young inequality applied for every point in the domain, in fact if $\| x \|_p = \|y\|_q = 1$ (any other case can be reduced to this normalizing the functions) then we have: $$ \sum \left| x_i y_i \right| \le \sum \left( \frac{\left| x_i \right|^p}{p} + \frac{ \left| y_i \right|^p}{q}\right) = \frac{\sum \left| x_i \right|^q}{p} + \frac{\sum \left| y_i \right|^q}{q} = \frac{1}{p} + \frac{1}{q} = 1 $$

The only inequality used is the Young inequality that it's an equality if and only if $$ \forall i \;\;\; \left| x_i\right|^p = \left| y_i\right|^q $$

This can be generalized to a generic measure space changing the $\forall i$ with a "almost for every $i$".

When does the equality hold in the Holder inequality?

1 Answers1