“Best possible” upper bounds for the first positive zeros of Bessel functions — the finite part

AbstractIt is shown here that the first three terms of the asymptotic expansion of jvk, k = 1, 2, 3, provide an upper bound for jvk in 0 < v ⩽ 10, hence a “best possible” upper bound. Lang and Wong have shown that this is true also for 10 < v < ∞ when k = 1 and 2, so that these “best possible” upper bounds hold in the entire interval 0 < v < ∞ in these cases. We include supplementary comments on lower bounds in 0 ⩽ v ⩽ 10