Monotonicity of the zeros of the third derivative of Bessel functions

Abstract

ABSTRACT. It is shown that j" f k is an increasing function of z/, 1 < v < oo, for each fixed k = 1,2,... , and also that this holds in 0 < v < oo when j"^ > y/S. Here j"' k is the k-th. positive zero of J'J'fa), the third derivative of the Bessel function of first kind and order v. These results follow from a representation derived for dj f J f k /du, u > 0. In addition, a number of inequalities for j"^ are established, especially for k = 1.

The zeros of the third derivative of Bessel functions of order less than one

“…This is by way of completing results found in [7], albeit by an entirely different method. There it was shown that all other positive zeros of J'J'fa), v > 0, are increasing functions of v.…”

“…In addition, as in [7], some inequalities for these zeros are established, here for i/o < i/ < 1 (cf. §5).…”

“…Acknowledgments Alfred Gray, as mentioned in [7], constructed graphics using Mathematica which depict the monotonicity properties established both in [7] and here. His diagrammes led to the conjectures established in §4 here.…”

Starlikeness of Bessel functions and their derivatives

“…ν,n − z 2 . On the other hand, we know that ν → j ′′′ ν,n is increasing on (2, ∞) for each n ∈ N fixed (see [14,17]), and thus the function ∞). Consequently, we have that the inequality…”