An upper bound for the zeros of the derivative of Bessel functions
type: Item
instance of: scholarly article
title: An upper bound for the zeros of the derivative of Bessel functions
zbMATH Open document ID: 0880.33003
DOI: 10.1007/BF02844477
author: Á. Elbert
author: A. Laforgia
publication date: 1998-02-08T00:00:00Z
review text: The aim of this paper is to establish a new upper bound for the positive zeros \(j_{\nu k}'\) contained in the following theorem: Let \(j_{\nu k}'\) denote the \(k\)-th positive zero of the derivative of Bessel functions \((d/dx)J_\nu(x)= J_\nu'(x)\) for \(\nu>0\), and \(k=1,2,\dots\;\). Then \(j_{\nu k}'<F_k(\nu)\), where \[ F_k(\nu)= \nu+a_k(\nu+A_k^3/a_k^3)^{1/3}+ {\textstyle\frac{3}{10}} a_k^2(\nu+A_k^3/a_k^3)^{-1/3}, \] and \[ A_k= {\textstyle\frac23} a_k\sqrt{2a_k}, \qquad a_k=2^{-1/3}x_k' \] and \(x_k'\) is the \(k\)-th positive zero of the derivative \(A_i'(x)\) of the Airy function. The bound is sharp for large values of \(\nu\) and improves known results. A similar inequality holds for the \(k\)-th positive zero \(y_{\nu k}'\) of \(Y_\nu'(x)\), too, where \(Y_\nu(x)\) denotes the Bessel function of the second kind.
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 1036021
zbMATH Keywords: Bessel functions
zbMATH Keywords: asymptotics
zbMATH Keywords: inequalities
zbMATH Keywords: zeros
zbMATH Keywords: Airy function
reviewed by: Som Prakash Goyal
MaRDI profile type: Publication
cites work: Concerning the Zeros of Some Functions Related to Bessel Functions
cites work: Q4364417
cites work: Q4122196
cites work: Non-existence of complex and purely imaginary zeros of a transcendental equation involving Bessel functions
cites work: The asymptotic solution of linear differential equations of the second order for large values of a parameter
cites work: Q5846799
full work available at URL: https://doi.org/10.1007/bf02844477
OpenAlex ID: W2046981736
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1036021