I am carrying out sum proof of a particular calculation and I am stuck at the following step. Let there be two functions of variable $\delta$ given by
$$f(\delta) = \left|\sum_{i=1}^N\frac{e^{j\pi i(\beta + \delta) \sin\theta} - e^{j \pi i (\beta - \delta) \sin\theta}}{2 j \pi i \beta \sin \theta} \right|, \qquad\text{and}\qquad g(\delta) = \left|\sum_{i=1}^N e^{j \pi i \delta \sin\theta} \right|.$$
where $1 \gg \beta > \delta > 0$ and $\pi \geq \theta \geq 0$ and $N$ is a large number in orders of hundreds. Here $\beta$, $\theta$ and $N$ are constants.
I need to prove that:
$$f \geq g$$
Now I carried out simulations to verify it but need some suggestion on how to proceed to prove for the step mathematically.
Let me also when mention the precoding step to see if there can be any altenative steps that could have been taken:
$f(\delta) = \left|\sum_{i=1}^N e^{j \pi i \delta \sin\theta} sinc\left(i \beta sin \theta \right) \right|$ and $g(\delta) = \left|\sum_{i=1}^N e^{j \pi i \delta \sin\theta} \right|$.
I am looking to prove $f(\delta) \geq g(\delta)$.