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The zeros of the third derivative of Bessel functions of order less than one. (English) Zbl 0843.33001
Main results: “If ${\lambda }_{1}$, ${\lambda }_{2}$ are zeros of ${J}_{\nu }^{\text{'}\text{'}\text{'}}\left(x\right)$, $0\le \nu <1$, ${\lambda }_{1}<{\lambda }_{2}<{j}_{\nu 1}$, then ${\lambda }_{1}$ is a steadily decreasing function of $\nu$ as $\nu$ increases to 1 and ${\lambda }_{2}$ is a steadily increasing function of $\nu$ as $\nu$ increases to 1. Further, there exists a (unique) value of $\nu ={\nu }_{0}$ such that ${J}_{\nu }^{\text{'}\text{'}\text{'}}\left(x\right)$ has two zeros when ${\nu }_{0}<\nu <1$ and none when $0<\nu <{\nu }_{0}$. When $\nu ={\nu }_{0}$, ${J}_{\nu }^{\text{'}\text{'}\text{'}}\left(x\right)$ has a douoble zero in $0;” “Let ${\lambda }_{1}\left(\nu \right)<{\lambda }_{2}\left(\nu \right)$ denote the zeros of ${J}_{\nu }^{\text{'}\text{'}\text{'}}\left(x\right)$ in $0, ${\nu }_{0}<\nu <1$. Then ${\lambda }_{1}\left(\nu \right)↓0$, ${\lambda }_{2}\left(\nu \right)↑\sqrt{3}={j}_{11}^{\text{'}\text{'}\text{'}}$ as $\nu ↑1$” (Theorems 4.1 and 4.2 in the paper, respectively). Some inequalities for the zeros studied are also established. Here ${\nu }_{0}=0·755578\cdots$; ${j}_{\nu 1}$ is, as usual, the first positive zero of the Bessel function ${J}_{\nu }\left(x\right)$ and ${j}_{\nu k}^{\text{'}\text{'}\text{'}}$ $\left(k=1,2,\cdots \right)$ stands for the positive zeros of the third derivative ${J}_{\nu }^{\text{'}\text{'}\text{'}}\left(x\right)$ and ${J}_{\nu }\left(x\right)$. The above results complete other ones found in another paper by the author and P. Szegö [Methods Appl. Anal. 2, 103-111 (1995; Zbl 0833.33003)].
##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$
##### Keywords:
monotonicity; inequalities; Bessel function; zeros