Lorch, Lee The zeros of the third derivative of Bessel functions of order less than one. (English) Zbl 0843.33001 Methods Appl. Anal. 2, No. 2, 147-159 (1995). Main results: “If \(\lambda_1\), \(\lambda_2\) are zeros of \(J'''_\nu(x)\), \(0\leq \nu< 1\), \(\lambda_1< \lambda_2< j_{\nu 1}\), 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 \(J'''_\nu(x)\) has two zeros when \(\nu_0< \nu< 1\) and none when \(0< \nu< \nu_0\). When \(\nu= \nu_0\), \(J'''_\nu(x)\) has a douoble zero in \(0< x< j_{\nu 1}\);” “Let \(\lambda_1 (\nu)< \lambda_2 (\nu)\) denote the zeros of \(J'''_\nu(x)\) in \(0< x< j_{\nu 1}\), \(\nu_0< \nu< 1\). Then \(\lambda_1 (\nu) \downarrow 0\), \(\lambda_2 (\nu) \uparrow \sqrt {3}= j'''_{11}\) as \(\nu \uparrow 1\)” (Theorems 4.1 and 4.2 in the paper, respectively). Some inequalities for the zeros studied are also established. Here \(\nu_0= 0.755578\dots\); \(j_{\nu 1}\) is, as usual, the first positive zero of the Bessel function \(J_\nu (x)\) and \(j'''_{\nu k}\) \((k= 1, 2, \dots)\) stands for the positive zeros of the third derivative \(J'''_\nu(x)\) and \(J_\nu(x)\). 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)]. Reviewer: N.Hayek (La Laguna) Cited in 1 Document MSC: 33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\) Keywords:monotonicity; inequalities; Bessel function; zeros Citations:Zbl 0833.33003 PDFBibTeX XMLCite \textit{L. Lorch}, Methods Appl. Anal. 2, No. 2, 147--159 (1995; Zbl 0843.33001) Full Text: DOI Digital Library of Mathematical Functions: §10.21(iv) Monotonicity Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.21(iv) Monotonicity Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions