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Zbl 0938.33002
Landau, L.J.
Ratios of Bessel functions and roots of $\alpha J_v(x)+xJ_v'(x)=0$. (English)
[J] J. Math. Anal. Appl. 240, No.1, 174-204 (1999). ISSN 0022-247X

The author shows that $\zeta J_\nu(\nu \zeta)/J_{\nu+1} (\nu \zeta)$ is stricty decreasing in $\nu$ on any interval not containing a singularity and obtaining an upper bound on its derivative: $${\partial \over \partial\nu} \left[{J_\nu (\nu\zeta)\over J_{\nu+1} (\nu\zeta)} \right]\le-{2 \over \nu^2}.$$ He also shows that $x{\partial\over \partial v}[{J\nu(x) \over J_{\nu+1} (x)}]\ge 2$ where $xJ_\nu(x)/J \nu+1(x)$ is strictly increasing in $\nu$. The graph of $J_\nu(x)/J_{\nu+1}(x)$ as a function of $x$ is illustrated in figures for various values of $\nu$. These results generalize and sharpen previously known results, and allow the deduction of a more complete description, including monotonicity and multiplicity, of the positive roots of the equation $\alpha J\nu(x)+xJ_\nu'(x)=0$ for all real $\nu$ and $\alpha$.
[R.S.Dahiya (Ames)]

MSC 2000:: *33C10 Cylinder functions, etc.

Keywords: Bessel functions

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Zentralblatt MATH Berlin [Germany]