Geodesic
Use of complex analysis for deriving lower bounds for trigonometric polynomials
Matematičeskie zametki, Tome 63 (1998) no. 6, pp. 803-811

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It is shown that for any distinct natural numbers $k_1,\dots,k_n$ and arbitrary real numbers $a_1,\dots,a_n$ the following inequality holds: $$ -\min_x\sum_{j=1}^na_j\bigl(\cos(k_jx)-\sin(k_jx)\bigr) \ge B\biggl(\frac 1{1+\ln n}\sum_{j=1}^na_j^2\biggr)^{1/2}, \qquad n\in\mathbb N, $$ where $B$ is a positive absolute constant (for example, $B=1/8$). An example shows that in this inequality the order with respect ton, i.e., the factor $(1+\ln n)^{-1/2}$, cannot be improved. A more elegant analog of Pichorides' inequality and some other lower bounds for trigonometric sums have been obtained. 
@article{MZM_1998_63_6_a0,
     author = {A. S. Belov},
     title = {Use of complex analysis for deriving lower bounds for trigonometric polynomials},
     journal = {Matemati\v{c}eskie zametki},
     pages = {803--811},
     publisher = {mathdoc},
     volume = {63},
     number = {6},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_63_6_a0/}
}
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A. S. Belov. Use of complex analysis for deriving lower bounds for trigonometric polynomials. Matematičeskie zametki, Tome 63 (1998) no. 6, pp. 803-811. http://geodesic.mathdoc.fr/item/MZM_1998_63_6_a0/