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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

$\left(1\right)\phantom{\rule{1.em}{0ex}}{u}^{\text{'}\text{'}}+\left(1+\frac{5}{36{\xi }^{2}}\right)u=0$

satisfied by (($\frac{3}{2}{\xi \right)}^{1/6}Ai\left[-{\left(\frac{3}{2}\xi \right)}^{2/3}\right]$ and ${\left(\frac{3}{2}\xi \right)}^{1/6}Bi\left[-{\left(\frac{3}{2}\xi \right)}^{2/3}\right]$ and the differential equation

$\left(2\right)\phantom{\rule{1.em}{0ex}}{v}^{\text{'}\text{'}}+\left[\frac{1}{2}\frac{{g}^{\text{'}\text{'}\text{'}}}{{g}^{\text{'}}}-\frac{3}{4}{\left(\frac{{g}^{\text{'}\text{'}}}{{g}^{\text{'}}}\right)}^{2}+{g}^{{}^{\text{'}}2}\right]v=0$

satisfied by $v\left(\xi \right)={\left({g}^{\text{'}}\right)}^{-1/2}cosg\left(\xi \right)$. Then the authors compare (1) and (2) when g($\xi$) 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.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory