I am following Drutu and Kapovich's Geometric Group Theory. Growth rates of functions are compared using the order given by $\asymp$ which is defined as:
Let $X\subseteq\mathbb{R}$. For functions $f, g: X \rightarrow \mathbb{R}$, write $f \preceq g$ if there exist $a,b>0$ and $x_0 \in \mathbb{R}$ such that for all $x\in X, x\geq x_0$, we have $bx\in X$ and $$f(x) \leq a g(bx)$$
Then $f \asymp g$ if and only if $f\preceq g$ and $g \preceq f$.
Question: If a function $f$ satisfies $x \preceq f \preceq x^\alpha$ for some $\alpha > 1$ with $\alpha \in \mathbb{R}$, then does that mean $f \asymp x^\beta$ for some $1\leq \beta \leq \alpha$?
If the answer is no, what if we are working with the growth function $f_G(n)$ of a finitely generated group $G$. In this case, does it have to be the case that $\beta$ is an integer?