Analogs of the Jackson–Nikol'skii inequalities for trigonometric polynomials in spaces with nonsymmetric norm

The paper is concerned with the evaluation of
$$ \sup_{\substack{t_n\in T_n\\t _n\not\equiv 0}} \frac{\|t_n\|_{q_1,q_2}}{\|t_n\|_{p_1,p_2}}, $$
where $\|\cdot\|_{p_1,p_2}$ is a nonsymmetric norm. The order of this number is obtained. Lower bounds involve new polynomials whose properties are studied in detail. In the case $p_1=p_2$, $q_1=q_2$, the estimate obtained is reduced to the well-known Jackson–Nikol'skii inequality.