Generalized Holder Inequality

Mathkk

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Let \(\displaystyle a_i \in \mathbb R^n\) with \(\displaystyle a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)\) for \(\displaystyle i = 1, ... , k\) and let \(\displaystyle p_1,...,p_k \in \mathbb R_{>1}\) with \(\displaystyle \frac1{p_1}+ ... + \frac1{p_k} = 1\)

Then show the following inequality by assuming that there are for every \(\displaystyle i = 1, ... ,k\) one \(\displaystyle N \in \mathbb N_{>1}\) and one \(\displaystyle n_i \in \{1,...,2^N\}\) with \(\displaystyle p_i = 2^N / n_i\):

\(\displaystyle \sum_{j=1}^n|\prod_{i=1}^ka_{i}^j| \leq \prod_{i=1}^k(\sum_{j=1}^n|a_{i}^j|^{p_i})^{1 \over {p_i}}\)

I am pretty much lost...
 
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Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$

Then show the following inequality by assuming that there are for every $i = 1, ... ,k$ one $N \in \mathbb N_{>1}$ and one $n_i \in \{1,...,2^N\}$ with $p_i = 2^N / n_i$:

$$\sum_{j=1}^n|\prod_{i=1}^ka_{i}^j| \leq \prod_{i=1}^k(\sum_{j=1}^n|a_{i}^j|^{p_i})^{1 \over {p_i}}$$

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Let \(\displaystyle a_i \in \mathbb R^n\) with \(\displaystyle a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)\) for \(\displaystyle i = 1, ... , k\) and let \(\displaystyle p_1,...,p_k \in \mathbb R_{>1}\) with \(\displaystyle \frac1{p_1}+ ... + \frac1{p_k} = 1\)

Then show the following inequality by assuming that there are for every \(\displaystyle i = 1, ... ,k\) one \(\displaystyle N \in \mathbb N_{>1}\) and one \(\displaystyle n_i \in \{1,...,2^N\}\) with \(\displaystyle p_i = 2^N / n_i\):

\(\displaystyle \sum_{j=1}^n|\prod_{i=1}^ka_{i}^j| \leq \prod_{i=1}^k(\sum_{j=1}^n|a_{i}^j|^{p_i})^{1 \over {p_i}}\)
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