Bounds for zeros of some special functions

Abstract:

For $n \geqq 1$ let ${b_n}$ and ${c_n}$ be zeros (ordered by increasing values) of $u(x)$ and $v(x)$, respectively, which are non-trivial solutions of $u'' + p(x)u = 0$ and $v'' + q(x)v = 0$ with continuous $p(x)$ and $q(x)$. It is shown that if ${b_n} - {c_n} \to 0$ as $n \to \infty ,\;p(x) \geqq q(x)$, and either $p(x)$ or $q(x)$ is nonincreasing, then ${b_n} \geqq {c_n}$ for $n \geqq 1$. Inequalities related to asymptotic expansions are obtained for the negative zeros ${a_n}$ of the Airy function $Ai(z)$ and the zeros ${j_{v,n}}$ of the Bessel function ${J_v}(x)$.