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By a generalized trigonometric polynomial, I shall mean a function $f:\mathbb R^+ \rightarrow \mathbb R$ given by an expression of the form
$$f(x) := \sum_{j=1}^k a_j \cos(\alpha_j x) + b_j \sin(\alpha_j x)$$ where $a_j, \alpha_j$ and $b_j$ are arbitrary real numbers such that $\{\alpha_j\}_{j=1}^k$ are all distinct. We assume that $f$ is non-periodic.

I am interested in finding useful absolute lower bounds on $|f'(r)|$ at the roots $r$ of $f$, which are valid uniformly for all sufficiently large zeroes $r$ (and in particular, are independent of $r$ itself).

Plotting a few trigonometric polynomials suggested that such an absolute lower bound should hold true, however I am not sure how to show such an inequality for $k>1$. I feel like there should be some result in literature on this, however despite an extensive search, I have not found any. Some attempts at using the Cauchy-Schwarz inequality have not led anywhere either. I would really appreciate any suggestions or references. Thank you.

Edit: After Oleg Eroshkin's counterexample, I have modified the question; hopefully, it excludes trivial counterexamples now.

Edit 2: If nothing else, it would also be quite interesting to know something about a lower bound on the average size of the absolute derivative of $f$ at the zeroes, something like a lower bound on $$\liminf_{X \rightarrow \infty} \frac1X \sum_{x_n \leq X} |f'(x_n)|^c$$ for some fixed $c>1$, where $x_1 < x_2 < \cdots$ are the zeroes of $f$ in ascending order.

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  • $\begingroup$ Unless I misunderstood the question, $f'(x_0)$ can be zero. This is just linear algebra: fix any $\alpha_j$ and $x_0$. You got a linear map from $(1_1, b_1,\dots,a_k,b_k)$ to $(f(x_0),f'(x_0))$. If $k>1$, you may find nontrivial coefficients $a_1, b_1,\dots$, such that $f(x_0)=f'(x_0)=0$. $\endgroup$
    – Oleg Eroshkin
    Commented Jan 15, 2022 at 3:29
  • $\begingroup$ @Oleg Eroshkin You are right, that was silly. I have modified the question, could you please take a look at it now? $\endgroup$
    – asrxiiviii
    Commented Jan 15, 2022 at 4:31
  • $\begingroup$ What kind of estimates you need? It's actually not clear to me that the limit must be non-zero. $\endgroup$
    – Oleg Eroshkin
    Commented Jan 15, 2022 at 19:51
  • $\begingroup$ @Oleg Eroshkin In Edit 2, I was looking for a positive lower bound depending only on $g$, say on $k, a_j, b_j, \alpha_j$ or other constants depending only on $g$ (say, its global extremum etc.). But a non-trivial lower bound depending at most on $g$ that is valid for any generalized trigonometric polynomial $g$. $\endgroup$
    – asrxiiviii
    Commented Jan 15, 2022 at 23:49

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