On the zeros of quotients of Bessel functions
The modified Bessel functions of the first kind are usually denoted by \(I_\nu(z)=e^{-i\pi\nu/2}J_\nu(z e^{i\pi/2})\). The authors prove that for \(m\geq 2\) the equation \[ {{I_m(x)}\over{I_{m-1}(x)}}- {{m+1}\over{2m}} {{I_1(x)}\over{I_0(x)}}= 0, \quad x\in{\mathbb R}_+, \] has a unique solution \(x_m\). Further on if \(2\leq k<m\), then \(x_k<x_m\) holds true. The main tool to prove this theorem is to show that the function \(I_1(x)/I_0(x)\) is concave on \({\mathbb R}_+\).
scientific article; zbMATH DE number 1530099
scientific article