On the power of the zeros of Bessel functions (Q1112992)

For \(\nu\geq 0\) let \(c_{\nu k}\) be the k-th positive zero of the cylinder function \[ C_{\nu}(t)=J_{\nu}(t)\cos \alpha - Y_{\nu}(t)\sin \alpha,\quad 0\leq \alpha <\pi, \] where \(J_{\nu}(t)\) and \(Y_{\nu}(t)\) denote the Bessel functions of the first and the second kind, respectively. We prove that \(c_{\nu k}^{1+H(x)}\) is convex as a function of \(\nu\), if \(c_{\nu k}\geq x>0\) and \(\nu\geq 0\), where H(x) is specified in Theorem 1.1.