Geodesic
Analogs of the Jackson--Nikol'skii inequalities for trigonometric polynomials in spaces with nonsymmetric norm
Matematičeskie zametki, Tome 61 (1997) no. 5, pp. 687-699

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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. 
@article{MZM_1997_61_5_a5,
     author = {A. I. Kozko},
     title = {Analogs of the {Jackson--Nikol'skii} inequalities for trigonometric polynomials in spaces with nonsymmetric norm},
     journal = {Matemati\v{c}eskie zametki},
     pages = {687--699},
     publisher = {mathdoc},
     volume = {61},
     number = {5},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_61_5_a5/}
}
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A. I. Kozko. Analogs of the Jackson--Nikol'skii inequalities for trigonometric polynomials in spaces with nonsymmetric norm. Matematičeskie zametki, Tome 61 (1997) no. 5, pp. 687-699. http://geodesic.mathdoc.fr/item/MZM_1997_61_5_a5/