On inflection points of Bessel functions of the second kind of positive order

ABSTRACT

In this paper, we study inflection points of Bessel functions of the second kind $Y_{\nu }\left(x\right)$ of positive order. In particular, we prove that there exists a positive number ${\nu }_{\ast }>0$ such that for all $\nu >{\nu }_{\ast }$ at least two positive inflection points reside before the first positive zero $y_{\nu }$ of $Y_{\nu }$. As a consequence, we prove that the function $\nu \mapsto y_{\nu }^{″}$, where $y_{\nu }^{″}$ is the smallest positive inflection point of $Y_{\nu }$, is discontinuous on $(0,\mathrm{\infty })$. We present lower and upper bounds for these inflection points and formulate some conjectures.

KEYWORDS:

• Bessel functions
• zeros
• inflection points

AMS SUBJECT CLASSIFICATION:

• 33C10
• 34A99

Acknowledgements

The authors are grateful to I.V. Tikhonov for stating the problem, and to A.Yu. Popov and V.B. Sherstyukov for their valuable remarks.

Disclosure statement

No potential conflict of interest was reported by the authors.