Lp-norm inequality for double sum

Question

I would need the following result

$\sum_{i=1}^N \sum_{t=1}^T |c_{it}| \leq (\sum_{i=1}^N \sum_{t=1}^T |c_{it}|^p)^{1/p} $

for $p<1$. I know there is an analogue for single sums, but does this hold for the double sum, and is there a reference I could cite?

Any help is very gratefully received.

Answer 1

Re-index

The $lp$ norm inequalities give:

$$\sum_{i=1}^M |a_{i}| \leq \left (\sum_{i=1}^M |a_{i}|^p\right)^{1/p} $$

You can use the same inequality. You have $M= N T$ many $c_{i,t}$ terms. So define a sequence:

$$a_1 = c_{1,1} \qquad a_2 = c_{2,1} \qquad \ldots \qquad a_N = c_{n,1}$$

$$a_{N+1} = c_{1,2} \qquad a_{N+2} = c_{2,2} \qquad \ldots \qquad a_{2N} = c_{N,2}$$

$$ \vdots $$

$$\ldots \qquad \ldots \qquad a_{TN} = c_{N,T}$$

and write

$$\sum_{i=1}^N \sum_{t=1}^T |c_{it}| = \sum_{i=1}^M |a_{i}| $$

Answer 2

It does not seem that a reference is needed: we have for non-negative $a_k$, $1\leqslant k\leqslant n$ and $0<p\leqslant 1$, $$\tag{*} \left(\sum_{k=1}^n a_k\right)^p\leqslant \sum_{k=1}^n a_k^p. $$ As a consequence, an application of $(*)$ with $n=N$, $a_i=\sum_{t=1}^T |c_{it}|$ gives $$ \left(\sum_{i=1}^N \sum_{t=1}^T |c_{it}|\right)^p\leqslant\sum_{i=1}^N \left(\sum_{t=1}^T |c_{it}|\right)^p $$ and an other application of $(*)$, this time with $n=T$ and $a_t= |c_{it}|$ gives the wanted conclusion.