Some consequences of a lower bound for the second derivative of the zeros of Bessel functions
For \(\kappa =1,2,...\), and \(\nu >-1\), let \(j_{\nu \kappa}\) be the \(\kappa\) th positive zero of \(J_{\nu}(x)\). In Analysis 5, 71-86 (1985; Zbl 0564.33005) and elsewhere, the authors extended this to continuous \(\kappa >0\) in such a way that for \(\nu >-\kappa\), k is the integer part of \(\kappa +1\), and \(\alpha =\pi (k-\kappa)\), \(j_{\nu \kappa}\) is the kth positive zero of the function cos \(\alpha\) \(J_{\nu}(x)-\sin \alpha Y_{\nu}(x)\). They showed (op. cit.) that \[ (\nu +j)j''>\nu j'{}^ 2/j- j',\quad for\quad \nu \geq 0 \] and \(\kappa \geq \kappa_ 0=0.7070...\), where \('=d/d\nu\). In the present paper they use this to prove that the following convex function of \(\nu\) : \(\log (j_{\nu \kappa}/\nu)\) \((\nu >0\), \(\kappa \geq \kappa_ 0)\); \(j^ 2_{\nu \kappa}\) (\(\nu\geq 0\), \(\kappa \geq \kappa_ 0)\); \(j_{\nu \kappa}/(\nu +1)\) (\(\nu\geq 0\), \(\kappa\geq 1)\).
scientific article; zbMATH DE number 4019464
scientific article