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An upper bound for the zeros of the derivative of Bessel functions. (English) Zbl 0880.33003
The aim of this paper is to establish a new upper bound for the positive zeros $j_{\nu k}'$ contained in the following theorem: Let $j_{\nu k}'$ denote the $k$-th positive zero of the derivative of Bessel functions $(d/dx)J_\nu(x)= J_\nu'(x)$ for $\nu>0$, and $k=1,2,\dots\ $. Then $j_{\nu k}'<F_k(\nu)$, where $$F_k(\nu)= \nu+a_k(\nu+A_k^3/a_k^3)^{1/3}+ {\textstyle\frac{3}{10}} a_k^2(\nu+A_k^3/a_k^3)^{-1/3},$$ and $$A_k= {\textstyle\frac23} a_k\sqrt{2a_k}, \qquad a_k=2^{-1/3}x_k'$$ and $x_k'$ is the $k$-th positive zero of the derivative $A_i'(x)$ of the Airy function. The bound is sharp for large values of $\nu$ and improves known results. A similar inequality holds for the $k$-th positive zero $y_{\nu k}'$ of $Y_\nu'(x)$, too, where $Y_\nu(x)$ denotes the Bessel function of the second kind.

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
Full Text: DOI
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