  For the sequence of multi-indexes $\{\alpha_i\}_{i=1}^{m}$ and $\beta$ we
study the inequality \[ \|D^{\beta} f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j=
1}^{m} \|D^{\alpha_j}f\|_{L_1(\mathbb{T}^d)}, \] where $f$ is a trigonometric
polynomial of degree at most $N$ on $d$-dimensional torus. Assuming some
natural geometric property of the set $\{\alpha_j\}\cup\{\beta\}$ we show that
\[ K_{N}\geq C \left(\ln N\right)^{\phi}, \] where $\phi<1$ depends only on the
set $\{\alpha_j\}\cup\{\beta\}$.
