# A Differential Equation for the Positive Zeros of the Function $\alpha J_v(z) + \gamma zJ_v'(z)$

### E.K. Ifantis

University of Patras, Greece### P.D. Siafarikas

University of Patras, Greece

## Abstract

A differential equation for any positive zero $\varrho(v)$ of the function $\alpha J_v(z) + \gamma zJ_v'(z)$ is found, where $J_v$ is the Bessel function of the first kind of $v > -1, J_v'$ 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 $\alpha \geq 1$, and the function $\varrho(v)/(\alpha + v)$ decreases with $v > -a$ in the case $\alpha < 1$.

(ii) The zeros of the function $\alpha J_v (z) + zJ_v'(z)$ increase with $v > -1$ in the case $\alpha \geq 1$ 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 $\alpha J_v(z) + \gamma zJ_v'(z)$ which complete and improve previously known bounds The second result improves a well-known result.

## Cite this article

E.K. Ifantis, P.D. Siafarikas, A Differential Equation for the Positive Zeros of the Function $\alpha J_v(z) + \gamma zJ_v'(z)$. Z. Anal. Anwend. 7 (1988), no. 2, pp. 185–192

DOI 10.4171/ZAA/295