An upper bound for the zeros of the derivative of Bessel functions

Abstract

Letj ^′ _vk denotes thekth positive zero of the derivativeJ ^′ _v (x)=dJ _v (x)/dx of Bessel functionJ _v (x) fork=1, 2,…. We establish the upper bound

$$j'_{\nu k}< \nu + a_k \left( {\nu + \frac{{{\rm A}_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} + \frac{3}{{10}}a_k^2 \left( {\nu + \frac{{A_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} , \nu \geqslant 0, k = 1,2, \ldots $$

whereA _k =2a _k √2a _k /3,a _k =x ^′ _k 2^−1/3 andx ^′ _k is thekth positive zero the derivativeAi′(x) of the Airy functionAi(x). This bound is sharp for large values ofv and improves known results. Similar inequality holds for thekth positive zeroy ^′ _vk ofY ^′ _v (x), too, whereY _v (x) denotes the Bessel function of second kind.