Best possible upper and lower bounds for the zeros of the Bessel function Jv(x)

Let j(nu, k) denote the k-th positive zero of the Bessel function J(nu)(x). In this paper, we prove that for nu > 0 and k = 1, 2, 3, ..., nu - a(k)/2(1/3) nu(1/3) < j(nu,k) < nu - a(k)/2(1/3) nu(1/3) + 3/20 a(k)(2) 2(1/3)/nu(1/3). These bounds coincide with the first few terms of the well-known asymptotic expansion j(nu,) (k) similar to nu - a(k)/2(1/3) nu(1/3) + 3/20a(k)(2)2(1/3)/nu(1/3) + ... as nu --> infinity, k being fixed, where a(k) is the k-th negative zero of the Airy function Ai(x), and so are "best possible".