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Inequalities for the zeros of the Airy functions. (English) Zbl 0722.33001

The authors use a comparison theorem due to Hethcote to establish asymptotic inequalities for the zeros of the Airy functions Ai(x) and Bi(x). They consider the differential equation

(1)u '' +(1+5 36ξ 2 )u=0

satisfied by ((3 2ξ) 1/6 Ai[-(3 2ξ) 2/3 ] and (3 2ξ) 1/6 Bi[-(3 2ξ) 2/3 ] and the differential equation

(2)v '' +[1 2g ''' g ' -3 4(g '' g ' ) 2 +g ' 2 ]v=0

satisfied by v(ξ)=(g ' ) -1/2 cosg(ξ). Then the authors compare (1) and (2) when g(ξ) is chosen in such a way that (1) and (2) are “very close”.

This reviewer observes that this idea is not original. Several applications of this idea have been given by L. Gatteschi [see, for example, L. Gatteschi, SIAM J. Math. Anal. 18, 1549-1562 (1987; Zbl 0639.33012)]. No mention is made by the authors of Gatteschi’s papers.

33C10Bessel and Airy functions, cylinder functions, 0 F 1
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory