On a method for generating inequalities for the zeros of certain functions

In this paper we describe a general procedure which yields inequalities satisfied by the zeros of a given function. The method requires the knowledge of a two-term approximation of the function with bound for the error term. The method was successfully applied many years ago [L. Gatteschi, On the zeros of certain functions with application to Bessel functions, Nederl. Akad. Wetensch. Proc. Ser. 55(3)(1952), Indag. Math. 14(1952) 224–229] and more recently too [L. Gatteschi and C. Giordano, Error bounds for McMahonʹs asymptotic approximations of the zeros of the Bessel functions, Integral Transform Special Functions, 10(2000) 41–56], to the zeros of the Bessel functions of the first kind. Here, we present the results of the application of the method to get inequalities satisfied by the zeros of the derivative of the function x J ν ( x ) . This function plays an important role in the asymptotic study of the stationary points of the solutions of certain differential equations.