The question is to find all harmonic functions $ u$ : $ R^{n} \rightarrow R $ satisfying $|u(x)| \leq C|x|^m \forall |x| \geq 1$ , where C is a constant and $ m \in (0,2)$
If m was an integer, I could have used derivative estimates for harmonic functions and concluded that the harmonic functions are polynomials of degree m. However, since in this case, $m \in (0,2)$ is any real number, I am not sure for to proceed.
Whether $m$ is an integer or not, those estimates you mention show that $$D^\alpha u(0)=0\quad(|\alpha|>m),$$hence $u$ is a polynomial of degree no larger than $m$. Since the degree is an integer it is no larger than $1$. Since any such polynomial is harmonic you're done. (If by chance $m<1$ then in fact $u$ is constant.)