Fractional derivatives and inequalities for trigonometric polynomials in spaces with asymmetric norms

We consider the Bernstein–Jackson–Nikol'skii inequalities for fractional derivatives in the case when the norm is asymmetric. Assume that $n\in\mathbb N$, $p_1,p_2,q_1,q_2\in[1,\infty]$, and $\alpha\in\mathbb R_+$. Then
$$ \sup_{\substack t_n\in\tau_n\\t_n\not\equiv 0}\dfrac{\|D^\alpha t_n\|_{q_1,q_2}}{\|t_n\|_{p_1,p_2}}\asymp I_\alpha n^{\alpha+\psi_1(p_1,p_2,q_1,q_2)}+n^{\alpha+\psi_2(p_1,p_2,q_1,q_2)}, $$
where
$$ I_\alpha=\begin{cases} \alpha,&0\leqslant\alpha\leqslant 1,\\ 1,&\alpha\geqslant 1, \end{cases} $$
and the functions $\psi_1$ and $\psi_2$ are given by an explicit formula. The asymptotic behaviour is with respect to $n$ for fixed $\alpha$, $p_1$, $p_2$, $q_1$ and $q_2$.