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

The generalized Airy function is a solution of the differential equation $(1)\quad y''+x^{\alpha}y=0,\quad x\in [0,\infty),$ 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 $$y(0)=0$$, 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 $$y(0)=0.$$”
Reviewer: L.Littlejohn

### MSC:

 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)

### Keywords:

monotonicity of zeros; Airy function
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### References:

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