About Moderately Decreasing functions

Question

A function $g: \mathbb{R} \to \mathbb{R} $ is said of moderate decrease if there is a $M \in \mathbb{R}$ such that $|g(x)| \leq \frac{M}{1 + |x|^{\alpha}}$, for some $\alpha > 1$, for all $x \in \mathbb{R}$. Supose $f$ is continuous and such that $f(x) = O(\frac{1}{x^\alpha})$ when $|x| \to + \infty$. Is $f$ of moderate decrease?

Answer 1

$(1+|x|^{\alpha} )|f(x)| \leq 2|x|^{\alpha} |f(x)|$ so $|f(x)| \leq \frac {2|x|^{\alpha} |f(x)|} {(1+|x|^{\alpha} )}$ for $|x| >1$. I hope this answers your question. (There are several errors in the statement; for example you have written 'for all $x \in \mathbb R$' in the definition which is not what you meant).