Bounds for the small real and purely imaginary zeros of Bessel and related functions

We give two distinct approaches to finding bounds, as functions of the order ν, for the smallest real or purely imaginary zero of Bessel and some related functions. One approach is based on an old method due to Euler, Rayleigh, and others for evaluating the real zeros of the Bessel function Jν(x) when ν > −1. Here, among other things, we extend this method to get bounds for the two purely imaginary zeros which arise in the case −2 < ν < −1. If we use the notation jν1 for the smallest positive zero, which approaches 0 as ν → −1, we can think of j ν1 as continued to −2 < ν < −1, where it has negative values. We find an infinite sequence of successively improving upper and lower bounds for j ν1 in this interval. Some of the weakest, but simplest, lower bounds in this sequence are given by 4(ν + 1) and 25/3(ν + 1)[(ν + 2)(ν + 3)]1/3 while a simple upper bound is 4(ν+1)(ν+2)1/2. The second method is based on the representation of Bessel functions as limits of Lommel polynomials. In this case, the bounds for the zeros are roots of polynomials whose coefficients are functions of ν. The earliest bounds found by this method already are quite sharp. Some are known in the literature though they are usually found by ad hoc methods. The same ideas are applied to get bounds for purely imaginary zeros of other functions such as J ′ ν(x), J ′′ ν (x), and αJν(x) + xJ ′ ν(x).