On Bernstein type quantitative estimates for Ornstein non-inequalities

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 $\{\alpha_j\}\cup\{\beta\}$.