We consider the discrete LlogL space of sequences $x=(x_i)$ such that 

$$\Vert x\Vert_{LL}:=\sum_i \vert x_i \log(x_i)\vert <\infty.$$

Let $x=(x_i)$ and $y=(y_i)$ two sequences in the above LlogL space.

Can we estimate for a sequence $ x_i \ge 0$

$$\Big \vert \sum_i x_i\log(x_i) \Big \vert \le \varepsilon \Vert x\Vert_{LL}+C_{\varepsilon} \Vert x\Vert^{1/2}_{\ell^1} $$