# Question about $|x^{k}\varphi^{(q)}(x)| \leqslant C_{q}A^{k}k^{k\alpha}$ in Gelfand Shilov spaces

I am reading into Gelfand Shilov spaces and in the book about distribution theory there is this space $$S_{\alpha}$$ defined by

$$|x^{k}\varphi^{(q)}(x)| \leqslant C_{q}A^{k}k^{k\alpha}$$

where $$C_{q}$$ and $$A$$ are constants and $$k,q = 0,1,2,\dots$$. Now according to the authors this definition imposes a constraint on the decrease of the fundamental functions as $$|x| \rightarrow \infty$$. They say that his is easily seen when dividing both sides by $$|x|^{k}$$ and pass to the minimum of $$k$$ on the right side. I don't really see that - when I do that I get

$$|\varphi^{(q)}(x)| \leqslant \frac{C_{q}A^{k}k^{k\alpha}}{|x|^{k}}$$

but what does passing to the minum mean? And how do I see the constraint?

Thanks!