An upper bound for the first zero of Bessel functions

It is shown, using the Rayleigh-Ritz method of the calculus of variations, that an upper bound for the first zeroy„, of z'v Jv(z), v > -1, is given by (,+ l)l/2{(„ + 2)l/2+l}, and that for large v,jr = v + 0(v]/2). 1. The following upper bound is given by Watson [4] for the first zero jv of Jv(x) (v>0) ¡A 1 '/2 (1) J. <{ f(" +!)(" + 5) j . It may be shown that a better bound may be obtained, valid for v > -1, namely (2) (,+ l)1/2((, + 2)1/2+l). 2. Consider the function (3) U(z) = T{v+\)(2/(yz))'j,(yz). The differential equation satisfied by u(z) is given by Watson [3] to be (4) z2u" + (2v + \)zu' + y2z2u = 0 with the boundary condition u(0) = 1, and if y is a zero of J„, u(\) — 0. Equation (4) can be written in Sturm-Liouville form (5) ¿(.-'fJ+TV-H.-* Multiplying Eq. (5) by u and integrating over 0 < z < 1, it follows that (6) 2=:/¿'2'+l"'2<fe U Y }¿z2'+xu2dx' On integration by parts, uu'z2v+x will vanish at 7 = 0, if v> -{, and u(\) vanishes. Thus the relation (6) provides a variational formulation, as indicated by Irving and Mullineux [1], for y2 which is an eigenvalue for the differential equation (5). The first eigenvalue will be7,2. The functional fl ,2v+l,,>2 j„ )q z u> az (7) A(«) = &z2>+Wdz Received February 19, 1981; revised July 13, 1981. 1980 Mathematics Subject Classification. Primary 33A40, 49G10. ©1982 American Mathematical Society 0025-5718/81/0000-1069/$01.50 589 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use