Further monotonicity and convexity properties of the zeros of cylinder functions (Q1200191)

If \(C_{\nu k}\) be the \(k\)th positive zero of the cylinder function \(C_ \nu(x,\alpha)=\cos(\alpha)J_ \nu(x)-\sin(\alpha)Y_ \nu(x)\), \(0\leq\alpha<\pi\) then it is shown that the function \(\nu({d^ 2(C_{\nu k}) \over d\nu^ 2}+\delta)/C_{\nu k}\) increases with \(\nu\geq 0\) for suitable values of \(\delta\) and \(k-\alpha/\pi\geq 0.7070\dots\), which is shown to imply under suitable conditions that \(C_{\nu k}+{1\over 2}\delta\nu^ 2\) is a convex function of \(\nu\geq 0\). Monotonicity properties of the function \(C_{\nu k}/\nu\) are also obtained. These results improve known results.