Proof for the bound of a complex exponential function

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 $$f(\delta) \geq g(\delta)$$.

• I am slightly confused by your question: a function takes one or more inputs, and spits out an output. What are the inputs? Is $f$ a function of $\delta$? $\beta$? $\theta$? $N$? Which quantities are constant, and which are variable? You also state that this is an intermediate step in some larger computation. It would be helpful if you could explain that larger computation a little (as this looks like it might be an xy problem). – Xander Henderson May 3 at 0:33
• Thanks, Xander for pointing out the mistake and I have edited the question accordingly. For the description of the problem I have, its difficult to explain the entire problem, however, I will try to give a sketch of what I am trying to do. So the variable $\delta$ is random variable and $g$ here is norm of inner product of a function $X$ of delta with a other function say $Y$, i.e $g = |<X,Y>|$. Whereas i found a expected function of $X$ say $\bar{X}$ and $f$ is $|<\bar{X},Y>|$. I am trying to prove $f \geq g$. – Chandan Pradhan May 3 at 2:05