Upper bounds for the first zeros of Bessel functions

Abstract

An upper bound for the first positive zero of the Bessel functions of first kind J μ( z) for μ > −1 is given. This upper bound is better than a number of upper bounds found recently by several authors. The upper bound given in this paper follows from a step of the Ritz's approximation method, applied to the eigenvalue problem of a compact self-adjoint operator, defined on an abstract separable Hilbert space. Some advantages of this method in comparison with other approximation methods are presented.