## On the square of the zeros of Bessel functions.(English)Zbl 0541.33001

For $$\nu \geq 0$$, let $$c\equiv c(\nu,k,\alpha)$$ be the kth positive x-zero of $J_{\nu}(x) \cos \alpha-Y_{\nu}(x) \sin \alpha,\quad 0\leq \alpha<\pi,$ and let $$c(\nu,k,\alpha)$$ be continued analytically to the $$\nu$$-interval $$(\alpha/\pi-k,0).$$ The authors’ main result, in a different notation, is that there exists $$\kappa_ 0$$, $$0<\kappa_ 0<1$$, such that when $$k-\alpha/\pi >\kappa_ 0$$ we have $$dc/d\nu>1$$ and $$d^ 2(c^ 2)/d\nu^ 2>0$$, $$0\leq \nu<\infty$$. This implies that $$j^ 2_{\nu k}$$ is convex on $$0\leq \nu<\infty$$ where $$j_{\nu k}=c(\nu,k,0)$$, $$k=1,2,..$$.. The authors’ method is to consider $$c(\nu,k,\alpha)$$ as the solution of $dc/d\nu =2c\int^{\infty}_{0}K_ 0(2c\quad \sinh \quad t)e^{-2\nu t}dt,$ which satisfies $$c\to 0$$ as $$\nu \downarrow(\alpha /\pi-k)$$. They show that $$j^ 2_{\nu k}$$ fails to be convex on $$(-k,\infty)$$ for $$k=2,3,..$$. but they conjecture that $$j^ 2_{\nu l}$$ is convex on $$(- 1,\infty)$$.
Reviewer: M.Muldoon

### MSC:

 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$

### Keywords:

zeros of Bessel functions; convex
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