Different Forms of Hölder's Inequality?

Question

Are the following forms of Hölder's Inequality same?

Is first simplification of second or rather a specific condition of second? if$1/p+1/q=1$ and $a_i,b_i,p,q >0$ then $$\sum_{i=1}^n{a_ib_i}\le \left(\sum^n_{i=1}{a_i}^p \right)^{1/p} \left( \sum^n_{i=1}{b_i}^q\right)^{1/q}$$

and

$$\prod^m_{i=1}\left(\sum_{j=1}^na_{ij}\right)\ge \left(\sum_{j=1}^n\sqrt[m]{\prod_{i=1}^ma_{ij}}\right)^m$$ and other basic condition missing have usual requirements like all of the $a_{ij}\ge 0$ etc.

See at the bottom of first page for linked pdf.

Thank You

Answer 1

These are two different inequalities; they only agree when $p=q=m=2$ (where you square the $a_{ij}$'s to get the $a$'s and $b$'s). They would require different but similar proofs.