I am following Young's book, An introduction to nonharmonic fourier series, and I am having trouble with the following "obvious" estimates (Ch.1,Sec.10,Th.14):
where $\lvert \delta_n \rvert \leq L <\frac{1}{4}$, and the norm we are working with is $\lVert \cdot\rVert_2$ in $L^2[-\pi,\pi]$.
Analizyng the growth of the involved functions seems pretty difficult, since the zeros of their derivatives are quite tricky. I have tried a more elemental approach, comparing sizes of numerators and denominators, but I found trouble with inequalities of the form
$$\sin{ \pi \delta_n} \geq \sin{ \pi L}$$ or $$\cos{ \pi \delta_n} \leq \cos{ \pi L},$$
since sine is increasing and cosine is decreasing in $[0,\frac{\pi}{2}]$, so I guess a more subtle bound might be used.