JournalszaaVol. 7, No. 2pp. 185–192

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

  • E.K. Ifantis

    University of Patras, Greece
  • P.D. Siafarikas

    University of Patras, Greece
A Differential Equation for the Positive Zeros of the Function $\alpha J_v(z) + \gamma zJ_v'(z)$ cover
Download PDF

Abstract

A differential equation for any positive zero ϱ(v)\varrho(v) of the function αJv(z)+γzJv(z)\alpha J_v(z) + \gamma zJ_v'(z) is found, where JvJ_v is the Bessel function of the first kind of v>1,Jvv > -1, J_v' is the derivative of JvJ_v and α\alpha, γ\gamma are real numbers. It is proven that:

(i) The function ϱ(v)/(1+v)\varrho(v)/(1 + v) decreases with v>1v > -1 in the case α1\alpha \geq 1, and the function ϱ(v)/(α+v)\varrho(v)/(\alpha + v) decreases with v>av > -a in the case α<1\alpha < 1.

(ii) The zeros of the function αJv(z)+zJv(z)\alpha J_v (z) + zJ_v'(z) increase with v>1v > -1 in the case α1\alpha \geq 1 and with v>αv > \alpha in the case α<1\alpha < 1. The first result leads to a number of lower and upper bounds for the zeros of the function αJv(z)+γzJv(z)\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 αJv(z)+γzJv(z)\alpha J_v(z) + \gamma zJ_v'(z). Z. Anal. Anwend. 7 (1988), no. 2, pp. 185–192

DOI 10.4171/ZAA/295