Let $f(x)=\Re(\sum_{k=1}^n a_k e^{i\lambda_k x})$ for $0 < \lambda_1 < \lambda_2 < \dots < \lambda_n$ and some complex $a_1$, $a_2$, $\dots$, $a_n$. What is the best (in some sense) estimate for $\inf_{[-M,M]} f(x)$ for large $M$ (in particular, for $M=+\infty$). For example, is it true that $\inf f(x)\leq -C|a_1|$ for some absolute $C$?

The best estimate I was able to get contains $\sum |a_i|$ in denominator, but I would like to have the one which does not tend to zero when we add many new terms.