simple sum inequality I have an inequality for sums that I can't proof, although I know it is true.

Let $h_{ij} = h_{ji}$ a real $n\times n$ matrix, and $h_{ijk} = - h_{ikj}$ a real $n\times n\times n$ tensor with $$\sum_{i=1}^n h_{ii} =0, \quad \sum_{j=1}^n h_{jji} = 0, \, \forall \, 1 \leq i\leq n$$
Then there should follow $$\sum_k \Big( \sum_i h_{ii} \, h_{iik} \Big)^2 \leq \sum_j h_{jj}^2 \sum_{i,k} h_{iik}^2.$$

I haven't managed to show it and I would be very grateful for help!

Isn't this just an application of the Cauchy-Schwarz inequality in $l^2$?
Let $a = (h_{ii})_i$, $b = (h_{iik})_i$. We have $$\bigg(\sum_{i=1}^n a_i \, b_i \bigg)^2 = (a,b)^2_{l^2} \leq \|a\|_{l^2}^2 \, \|b\|_{l^2}^2 = \sum_{i=1}^n h_{ii}^2 \sum_{i=1}^n h_{iik}^2.$$
