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.
Best Answer
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 $i$".