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Monotonicity of the zeros of the third derivative of Bessel functions. (English) Zbl 0833.33003
The aim of the authors is to show that the zeros of the third derivative of the Bessel function $J_\nu (\lambda)$ are increasing functions of the order $\nu$. To this end they consider the boundary-value problem satisfied by $y(x) = J_\nu (\lambda x)$, namely: $$(xy')' = (\nu^2/x) y - \lambda^2 xy, \quad x \in (0,1) \qquad y(0) = 0, \quad y'''(1) = 0,$$ which permits them to show that $d\lambda/d \nu > 0$ when $\nu > 0$, $\lambda \ge \sqrt 3$ and $\lambda \ne j'''_{11}$. Here $j'''_{11}$ stands for the first zero of $j'''_{11}(\lambda)$. Notice that all the results obtained in the paper concern the case where $\nu$ is real (and positive).

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_0F_1$ 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators