A differential equation for the positive zeros of the function \(\alpha J_{\nu}(z)+\gamma zJ'_{\nu}(z)\)
Summary: A differential equation for any positive zero \(\rho(\nu)\) of the function \[ \alpha J_{\nu}(z) + \gamma zJ'_{\nu}(z) \] is found, where \(J_{\nu}\) is the Bessel function of the first kind of order \(\nu >-1\), \(J'_{\nu}\) is the derivative of \(J_{\nu}\) and \(\alpha\), \(\gamma\) are real numbers. It is proven that: (i) The function \(\rho(\nu)/(1+\nu)\) decreases with \(\nu >-1\) in the case \(\alpha\geq 1\), and the function \(\rho (\nu)/(\alpha +\nu)\) decreases with \(\nu >-\alpha\) in the case \(\alpha <1.\) (ii) The zeros of the function \(\alpha J_{\nu}(z)+zJ'_{\nu}(z)\) increase with \(\nu >-1\) in the case \(\alpha\geq 1\) and with \(\nu >-\alpha\) in the case \(\alpha <1.\) The first result leads to a number of lower and upper bounds for the zeros of the function \(\alpha J_{\nu}(z)+ zJ'_{\nu}(z)\) which complete and improve previously known bounds. The second result improves a well-known result.
scientific article; zbMATH DE number 4050002