Is this some kind of Holder's inequality? I am seeing this particular inequality for an array of real numbers $\{ x_{ij} \}$,
$$\sum_i \vert \prod_{j=1}^k x_{ij} \vert \leq \prod_{j=1}^k ( \sum_i \vert x_{ij}\vert ^k)^{\frac{1}{k}}$$ 
I would like to know the proof of this!
 A: Proof by induction on $k$. When $k = 1$ the inequality above is an equality. Now suppose $k > 1$ and the result holds for all positive integers less than $k$. By Hölder's inequality (with conjugate exponents $k$ and $k/(k-1)$), 
$$\sum_{i} \left\lvert \prod_{j = 1}^k x_{ij}\right\rvert  = \sum_i \left\lvert \prod_{j = 1}^{k-1} x_{ij}\right\rvert \lvert x_{ik}\rvert \le \left(\sum_i \left\lvert \prod_{j = 1}^{k-1} x_{ij}\right\rvert^{k/(k-1)}\right)^{(k-1)/k}\left(\sum_i \lvert x_{ik}\rvert^k\right)^{1/k}.$$
Now 
$$\left(\sum_i \left\lvert \prod_{j = 1}^{k-1} x_{ij}\right\rvert^{k/(k-1)}\right)^{(k-1)/k} = \left(\sum_i \left\lvert \prod_{j = 1}^{k-1} x_{ij}^{k/(k-1)}\right\rvert\right)^{(k-1)/k} \le \prod_{j = 1}^{k-1} \left(\sum_i \lvert x_{ij}^{k/(k-1)}\rvert^{k-1}\right)^{1/k},$$
using the induction hypothesis in the last step. Thus
$$\left(\sum_i \left\lvert \prod_{j = 1}^{k-1} x_{ij}\right\rvert^{k/(k-1)}\right)^{(k-1)/k} \le \sum_{j = 1}^{k-1} \left(\sum_i \lvert x_{ij} \rvert^k\right)^{1/k},$$
and consequently
$$\sum_i \left\lvert \prod_{j = 1}^k x_{ij}\right\rvert \le \prod_{j = 1}^{k-1} \left(\sum_i \lvert x_{ij}\rvert^k\right)^{1/k} \left(\sum_i \lvert x_{ik}\rvert^k\right)^{1/k} = \prod_{j = 1}^k \left(\sum_i \lvert x_{ij}\rvert^k\right)^{1/k},$$
as desired.
