A Differential Equation for the Positive Zeros of the Function $\alpha J_v(z)+\gamma z{J}_v^{\prime }(z)$

Abstract

A differential equation for any positive zero $\varrho (v)$ of the function $\alpha J_v(z)+\gamma z{J}_v^{\prime }(z)$ is found, where $J_v$ is the Bessel function of the first kind of $v>-1,J_v^{\prime }$ is the derivative of $J_v$ and $\alpha$, $\gamma$ are real numbers. It is proven that:

(i) The function $\varrho (v)/(1+v)$ decreases with $v>-1$ in the case $\alpha \ge 1$, and the function $\varrho (v)/(\alpha +v)$ decreases with $v>-a$ in the case $\alpha <1$.

(ii) The zeros of the function $\alpha J_v(z)+z{J}_v^{\prime }(z)$ increase with $v>-1$ in the case $\alpha \ge 1$ and with $v>\alpha$ in the case $\alpha <1$. The first result leads to a number of lower and upper bounds for the zeros of the function $\alpha J_v(z)+\gamma z{J}_v^{\prime }(z)$ which complete and improve previously known bounds The second result improves a well-known result.