Let $x \in \mathbb{R_{\geq 0}}$, $\alpha \in (0,1]$ and $p \in \mathbb{N}$. Define $$S(x)=\sum_{k=0}^{p-1} \frac{(e^{\frac{2\pi k}{p}i})^{x+1}}{e^{\frac{2\pi k}{p}i}-\alpha} $$
Does there exists $C>0$ such that $S(x) \geq C$ for all $x \in \mathbb{R_{\geq 0}}$?

I encountered similar summation in a journal article, but that was for a special case. I am trying to generalise the summation as above, if possible.

Note that the summation is always real because it involves summing over conjugate pairs. I tried splitting the sum into conjugate pairs and check whether each pair sums to a positive number. However, I am still not sure on how to proceed with that.

Is it true? May someone kindly provide proof for that? If not, is there any suitable modification on the assumption so that it holds true?


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