It is known that, for $z \gg \nu^2$, we have the asymptotic bound
$$\displaystyle |J_{\nu}(z)| \leqslant C_{\nu}|z|^{-1/2},$$
where $J_{\nu}$ denotes the Bessel function of the first kind, and $C_{\nu}$ denotes some positive constant depending on $\nu$. There are also different asymptotics for $z \ll 1$. I'd like to know if, for sufficiently large $z$, we can do better. For instance, suppose that $z \gg 10^{100}$. Can we do any better than the above bound, or is the above bound optimal? Can we take $C = 5000$ and the power of $|z|$ to be $-101/200$, for example?
No, the Bessel functions are oscillatory, but have an envelope decreasing in $1/\sqrt z$. This is very well confirmed by the half-integer case, which is exactly the ratio of a sinusoid over the square root.
The bound $\sqrt{\dfrac2{\pi z}}$ is tight.