Let $J_n$ be the Bessel function of the first kind. Let $J_n^{(\max)} = \max_{x>0} J_n(x)$. What is known about the asymptotic behavior of $J_n^{(\max)}$ at large $n$? Specifically, I am looking for a lower bound. It is ok if the result only holds for integer and half-integer $n$.
(It is potentially helpful to note that $J_n^{(\max)} = J_n(j'_n)$ where $j'_n$ is the smallest positive zero of $J'_n$, i.e. the global maximum of $J_n$ occurs at the "first" local maximum.)
