Let $J_{\nu}$ denote the Bessel function of the first kind. By considering the asymptotics of the Bessel function, as given here, we have that, as $z \rightarrow \infty,$ one has:
$$\displaystyle |J_{\nu}(z)| \leqslant C_{\nu}|z|^{-1/2},$$
where $C_{\nu}$ is some constant depending on $\nu$. My question is simple: is there a similar lower bound for $J_{\nu}$? I've searched for some bounds online, but none seem to give a lower bound for $J_{\nu}$. Instead, they give lower bounds for products or quotients of Bessel functions. Does anyone know of a lower bound, preferably one that consists of powers of $|z|$, as $z \rightarrow \infty$?
For some context, I'm trying to find a lower bound so that I can bound a particular sum from below. I've found identities involving modified Bessel functions, but they all seem to give bounds on $\frac{I_{\nu}(x)}{I_{\nu}(y)}$.