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Zbl 0722.33001
Pittaluga, Giovanna; Sacripante, Laura
Inequalities for the zeros of the Airy functions. (English)
[J] SIAM J. Math. Anal. 22, No.1, 260-267 (1991). ISSN 0036-1410; ISSN 1095-7154/e

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 $$ (1)\quad u''+(1+\frac{5}{36\xi\sp 2})u=0 $$ satisfied by (($\frac{3}{2}\xi)\sp{1/6}Ai[-(\frac{3}{2}\xi)\sp{2/3}]$ and $(\frac{3}{2}\xi)\sp{1/6}Bi[-(\frac{3}{2}\xi)\sp{2/3}]$ and the differential equation $$ (2)\quad v''+[\frac{1}{2}\frac{g'''}{g'}- \frac{3}{4}(\frac{g''}{g'})\sp 2+g\sp{'2}]v=0 $$ satisfied by $v(\xi)=(g')\sp{-1/2} \cos g(\xi)$. Then the authors compare (1) and (2) when g($\xi$) is chosen in such a way that (1) and (2) are ``very close''. \par This reviewer observes that this idea is not original. Several applications of this idea have been given by L. Gatteschi [see, for example, {\it L. Gatteschi}, SIAM J. Math. Anal. 18, 1549-1562 (1987; Zbl 0639.33012)]. No mention is made by the authors of Gatteschi's papers.
[A.Laforgia (L'Aquila)]

MSC 2000:: *33C10 Cylinder functions, etc.
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: zeros of Airy functions; asymptotic expansions; comparison theorem

Citations: Zbl 0639.33012

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