An Inequality Related the Zeros of Two Ordinary Bessel Functions

Abstract

AMS(MOS): 33A40, 47B05

Let v be real, v>-1 and v>Reμ, μ∊⊄ We prove an inequality which relates the first positive zero of the ordinary Bessel function J_v(z) and the absolute value of the real part of any zero of J_μ(z). Some lower bounds for the absolute value of the complex zeros of J_μ(z) follow immediately. In particular for μ real and μ>-1, this inequality proves that (1+v)^-1ρ_v,1 is a strictly decreasing function in the interval -1<v<+∞ where ρ_v,1 is the first positive zero of J_v(z). A number of simple lower and upper bounds for the first positive zero of J_v(z) follow immediatly from this result.

Additional information

Notes on contributors

D. Colton

E. K. Ifaktis

P. D. Siafarikas