Abstract:
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.
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