Bounds for zeros of some special functions

Abstract

For n> 1 let bn and c. be zeros (ordered by in- creasing values) of u(x) and v(x), respectively, which are non- trivial solutions of u+p(x)u=O and v+q(x)v=O with contin- uous p(x) and q(x). It is shown that if bn-cc---O as n- oo, p(x) > q(x), and either p(x) or q(x) is nonincreasing, then bn _cn for n _ 1. Inequalities related to asymptotic expansions are obtained for the negative zeros an of the Airy function Ai(z) and the zeros j,,n of the Bessel function J.(x). The principal theorem, which gives an inequality for zeros of solu- tions of linear second order differential equations, is proved by induc- tion using the Sturm comparison theorem at each step. This proce- dure differs from previous applications of the Sturm comparison theo- rem to orthogonal polynomials (3, pp. 120-130) and Bessel functions (4, pp. 518-521 ) since the common zero of the solutions of the differ- ential equations is at infinity here, i.e., approached asymptotically. THEOREM 1. For n > 1 let bn and cn be zeros (ordered by increasing values) of u(x) and v(x), respectively, which are nontrivial solutions of u+p(x)u=O and v+q(x)v=O with continuous p(x) and q(x). If bn - Cn>O as n- oo0, p(x) _q(x), and either p(x) or q(x) is nonincreas- ing, then bn _cCn for n _ 1. PROOF. The zeros bn and Cn are simple and cannot have a finite accumulation point (1, pp. 223-225). For p(x)=_q(x), the theorem is true since bn=cn. Assume p(x) 4q(x) and bn 0. We will show by induction that dn _ dk > 0 for all n > k. Assume dm.dk > 0 for some m > k. If = v(x+dm), then w(x) +q(x+dm)w(x) =0, w(bm) =0, and w(cm+i-dm) ==0. Since either p(x) >p(x+dm) > q(x+dm) or p(x) _ q(x) _ q(x+d.), the Sturm comparison theorem implies bm dm > dk>0, which completes our induction proof. However, cn-bn=dn> dk>O for all n> k contradicts bn-Cn--*0. Consequently, no least positive