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 $$\mathrm{\parallel }{D}^{\beta }f{\mathrm{\parallel }}_{L_1({\mathbb{T}}^d)}\le K_N\sum _{j=1}^m\mathrm{\parallel }{D}^{{\alpha }_j}f{\mathrm{\parallel }}_{L_1({\mathbb{T}}^d)},$$ 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 $$K_N\ge C{(\ln N)}^{\phi },$$ where $\phi <1$ depends only on the set $\{{\alpha }_j\}\cup \{\beta \}$.