Use of complex analysis for deriving lower bounds for trigonometric polynomials

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.