# A Differential Equation for the Positive Zeros of the Function $αJ_{v}(z)+γzJ_{v}(z)$

### E.K. Ifantis

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

University of Patras, Greece

## Abstract

A differential equation for any positive zero $ϱ(v)$ of the function $αJ_{v}(z)+γzJ_{v}(z)$ 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 $α$, $γ$ are real numbers. It is proven that:

(i) The function $ϱ(v)/(1+v)$ decreases with $v>−1$ in the case $α≥1$, and the function $ϱ(v)/(α+v)$ decreases with $v>−a$ in the case $α<1$.

(ii) The zeros of the function $αJ_{v}(z)+zJ_{v}(z)$ increase with $v>−1$ in the case $α≥1$ and with $v>α$ in the case $α<1$. The first result leads to a number of lower and upper bounds for the zeros of the function $αJ_{v}(z)+γzJ_{v}(z)$ 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 $αJ_{v}(z)+γzJ_{v}(z)$. Z. Anal. Anwend. 7 (1988), no. 2, pp. 185–192

DOI 10.4171/ZAA/295