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Monotonicity properties of zeros of generalized Airy functions. (English) Zbl 0681.33008

The generalized Airy function is a solution of the differential equation

$\left(1\right)\phantom{\rule{1.em}{0ex}}{y}^{\text{'}\text{'}}+{x}^{\alpha }y=0,\phantom{\rule{1.em}{0ex}}x\in \left[0,\infty \right),$

where $\alpha$ is a positive number. From the introduction: “M. S. P. Eastham conjectured that the first positive zero ${a}_{\alpha 1}$ of a solution of (1) with $y\left(0\right)=0$, decreases as $\alpha$ increases. We show here that this decrease (to 1) occurs for all positive zeros of such a solution and indeed for all, except possibly the first of the zeros of any nontrivial solution of (1) even without the condition $y\left(0\right)=0·$

Reviewer: L.Littlejohn

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34B30 Special ODE (Mathieu, Hill, Bessel, etc.)
##### Keywords:
monotonicity of zeros; Airy function
##### References:
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