# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
An upper bound for the zeros of the derivative of Bessel functions. (English) Zbl 0880.33003

The aim of this paper is to establish a new upper bound for the positive zeros ${j}_{\nu k}^{\text{'}}$ contained in the following theorem: Let ${j}_{\nu k}^{\text{'}}$ denote the $k$-th positive zero of the derivative of Bessel functions $\left(d/dx\right){J}_{\nu }\left(x\right)={J}_{\nu }^{\text{'}}\left(x\right)$ for $\nu >0$, and $k=1,2,\cdots \phantom{\rule{4pt}{0ex}}$. Then ${j}_{\nu k}^{\text{'}}<{F}_{k}\left(\nu \right)$, where

${F}_{k}\left(\nu \right)=\nu +{a}_{k}{\left(\nu +{A}_{k}^{3}/{a}_{k}^{3}\right)}^{1/3}+\frac{3}{10}{a}_{k}^{2}{\left(\nu +{A}_{k}^{3}/{a}_{k}^{3}\right)}^{-1/3},$

and

${A}_{k}=\frac{2}{3}{a}_{k}\sqrt{2{a}_{k}},\phantom{\rule{2.em}{0ex}}{a}_{k}={2}^{-1/3}{x}_{k}^{\text{'}}$

and ${x}_{k}^{\text{'}}$ is the $k$-th positive zero of the derivative ${A}_{i}^{\text{'}}\left(x\right)$ 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 ${y}_{\nu k}^{\text{'}}$ of ${Y}_{\nu }^{\text{'}}\left(x\right)$, too, where ${Y}_{\nu }\left(x\right)$ denotes the Bessel function of the second kind.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$
##### Keywords:
Bessel functions; asymptotics; inequalities; zeros; Airy function
##### References:
 [1] Abramowitz M., Stegun I. A.,Handbook of mathematical functions, Dover, New York, 1978. [2] Boyer T. H.,Concerning the zeros of some functions related to Bessel function, J. Math. Phys.,10 (1969), 1729–1744. · Zbl 0187.01402 · doi:10.1063/1.1665021 [3] Elbert Á., Laforgia A.,A lower bound for the zeros of the Bessel functions, WSSIAA, (1944), 179–185. [4] Gatteschi L., Laforgia A.,Nuove disuguaglianze per il primo zero ed il primo massimo della funzione di Bessel J v (x), Rend. Sem; Mat. Univ. Pol. Torino,34 (1975), 411–424. [5] Kokologiannaki C. G., Siafarikas P. D.,Non-existence of complex and purely imaginary zeros of a transcendetal equation involving Bessel functions, Zeitschrift für Analysis und ihre Anwendungen,10 (1991), 563–567. [6] Olver F. W. J.,The asymptotic expansion of Bessel function of large order, Philos. Trans. Roy. Soc. London Ser. A,247 (1954), 328–368. · doi:10.1098/rsta.1954.0021 [7] Watson G. N.,A treatise on the theory of Bessel functions (Second Ed.), Cambridge University Press, London, New York, 1944.