zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Monotonic sequences related to zeros of Bessel functions. (English) Zbl 1171.33304
Some inequalities concerning the values of the Bessel functions are proved by A. Akiyama and Y. Tanigawa in the course of their work on Salem numbers and uniform distribution modulo 1. This raises the question of inequalities and monotonicity properties for the sequences of values of one cylinder function at the zeros of another such function. Such results by differential equations methods are derived in this paper.

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
Full Text: DOI
[1] Akiyama, S., Tanigawa, Y.: Salem numbers and uniform distribution modulo 1. Publ. Math. (Debr.) 64, 329--341 (2004) · Zbl 1072.11053
[2] Elbert, Á.: An approximation for the zeros of Bessel functions. Numer. Math. 59, 647--657 (1991) · Zbl 0760.65020 · doi:10.1007/BF01385801
[3] Elbert, Á.: Some recent results on the zeros of Bessel functions and orthogonal polynomials. Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999). J. Comput. Appl. Math. 133(1--2), 65--83 (2001) · Zbl 0989.33004 · doi:10.1016/S0377-0427(00)00635-X
[4] Elbert, Á., Gatteschi, L., Laforgia, A.: On the concavity of zeros of Bessel functions. Appl. Anal. 16, 261--278 (1983) · Zbl 0512.33003 · doi:10.1080/00036818308839474
[5] Elbert, Á., Laforgia, A.: On the square of the zeros of Bessel functions. SIAM J. Math. Anal. 15, 206--212 (1984) · Zbl 0541.33001 · doi:10.1137/0515017
[6] Elbert, Á., Laforgia, A.: Monotonicity properties of the zeros of Bessel functions. SIAM J. Math. Anal. 17, 1483--1488 (1986) · Zbl 0597.33007 · doi:10.1137/0517106
[7] Elbert, Á., Laforgia, A.: Further results on McMahon’s asymptotic approximations. J. Phys. A: Math. Gen. 33, 6333--6341 (2000) · Zbl 0962.33005 · doi:10.1088/0305-4470/33/36/305
[8] Gatteschi, L.: Valutazione dell’errore nella formula di McMahon per gli zeri della J n (x) di Bessel nel caso 0 n 1. Rivista Mat. Univ. Parma 1, 347--362 (1950) · Zbl 0040.03103
[9] Gatteschi, L.: Funzioni Speciali. UTET, Torino (1973)
[10] Gatteschi, L.: Asymptotics and bounds for the zeros of Laguerre polynomials: A survey. J. Comput. Appl. Math. 144, 7--27 (2002) · Zbl 1008.65011 · doi:10.1016/S0377-0427(01)00549-0
[11] Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964) · Zbl 0125.32102
[12] Laforgia, A.: Sugli zeri delle funzioni di Bessel. Calcolo 17, 211--220 (1980) · Zbl 0464.33007 · doi:10.1007/BF02576701
[13] Lorch, L., Szego, P.: Higher monotonicity properties of certain Sturm-Liouville functions. Acta Math. 109, 55--73 (1963) · Zbl 0111.06502 · doi:10.1007/BF02391809
[14] Muldoon, M.E.: Continuous ranking of zeros of special functions. J. Math. Anal. Appl. (2008). doi: 10.1016/j.jmaa.2008.01.082 · Zbl 1134.33005
[15] Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944) · Zbl 0063.08184