Upper bound for $\left|(a+\pi)^k e^{i \pi x }- (a-\pi)^k e^{-i \pi x }\right|$.

Question

I want to find an upper bound for

$$\left|(a+\pi)^k e^{i \pi x }- (a-\pi)^k e^{-i \pi x }\right|\leq ?$$

where $a,k\in\mathbb{N}, x\in \mathbb{R}$.

thanks a lot

Answer 1

A very generous bound:$\left|(a+\pi)^k e^{i \pi x }- (a-\pi)^k e^{-i \pi x }\right|\leq|(a+\pi)^k e^{i \pi x }|+|(a-\pi)^k e^{i \pi x }|\leq |a+\pi|^k+|a-\pi|^k$