Skip to main content
SpringerLink
Log in

Studies on the zeros of Bessel functions and methods for their computation: 3. Some new works on monotonicity, convexity, and other properties

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

This paper continues the study of real zeros of Bessel functions begun in the previous parts of this work (see M. K. Kerimov, Comput. Math. Math. Phys. 54 (9), 1337–1388 (2014); 56 (7), 1175–1208 (2016)). Some new results regarding the monotonicity, convexity, concavity, and other properties of zeros are described. Additionally, the zeros of q-Bessel functions are investigated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. Abramowitz and I. N. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972; Nauka, Moscow, 1979).

    MATH  Google Scholar 

  2. R. A. Askey, “The q-gamma and q-beta-functions,” Appl. Anal. 8, 125–141 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  3. J. M. Blatt and S. T. Butler, “Superfluidity of an ideal Bose–Einstein gas,” Phys. Rev. 100, 476–480 (1995).

    Article  MathSciNet  Google Scholar 

  4. M. Bôcher, “On certain methods of Sturm and their application to the roots of Bessel functions,” Bull. Am. Math. Soc. 3, 205–213 (1897).

    Article  MATH  Google Scholar 

  5. A. Elbert, “Some inequalities concerning Bessel functions of firs kind,” Studia Sci. Math. Hungarica 6, 277–285 (1971).

    MathSciNet  MATH  Google Scholar 

  6. A. Elbert, “Concavity of the zeros of Bessel functions,” Studia Sci. Math. Hungarica 12, 81–88 (1977).

    MathSciNet  MATH  Google Scholar 

  7. A. Elbert, L. Gatteschi, and A. Laforgia, “On the concavity of zeros of Bessel functions,” Appl. Anal. 16 (4), 261–278 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Elbert and A. Laforgia, “On the zeros of derivatives of Bessel functions,” Z. Angew. Math. Phys. 34 (6), 774–786 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Elbert and A. Laforgia, “On the square of the zeros of Bessel functions,” SIAM J. Math. Anal. 15 (1), 206–212 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Elbert and A. Laforgia, “An asymptotic relation for the zeros of Bessel functions,” J. Math. Anal. Appl. 98, 502–511 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Elbert and A. Laforgia, “Further results on the zeros of Bessel functions,” Analysis 5, 71–86 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Elbert and A. Laforgia, “On the convexity of the zeros of Bessel functions,” SIAM J. Math. Anal. 16 (3), 614–619 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Elbert and A. Laforgia, “Monotonicity properties of the zeros of Bessel functions,” SIAM J. Math. Anal. 17 (6), 1483–1488 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Elbert and A. Laforgia, “Some consequences of a lower found for the second derivative of the zeros of Bessel functions,” J. Math. Anal. Appl. 125 (1), 1–5 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Elbert and A. Laforgia, “On the power of the zeros of Bessel functions,” Rend. Circ. Mat. Palermo 36 (2), 507–517 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Elbert and A. Laforgia, “A convexity property of zeros of Bessel functions,” J. Appl. Math. Phys. (ZAMP) 41 (5), 734–737 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Elbert, A. Laforgia, and L. Lorch, “Additional monotonicity properties of the zeros Bessel functions,” Analysis 11 (4), 293–299 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Elbert and P. D. Siafarikas, “On the square of the first zero of Bessel function J(z),” Can. Math. Bull. 20 (1), 1–12 (1998).

    MathSciNet  Google Scholar 

  19. C. Giordano and A. Laforgia, “Further properties of the zeros of Bessel functions,” Le Matematiche (Catania) 42, 19–28 (1987).

    MathSciNet  MATH  Google Scholar 

  20. C. Giordano and L. G. Rodono, “Further monotonicity and convexity properties of the zeros of cylinder functions,” J. Comput. Appl. Math. 42 (2), 245–251 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  21. W. Hahn, “Beitrage zur Theorie der Heineschen Reihen,” Math. Nachr. 2, 340–379 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Hurwitz, “Über die Nullstellen Besselschen Functionen,” Math. Anal. 33, 246–266 (1880).

    Article  MathSciNet  Google Scholar 

  23. E. K. Ifantis and P. D. Siafarikas, “A differential equations for the zeros of Bessel functions,” Appl. Anal. 20, 269–281 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  24. E. K. Ifantis and P. D. Siafarikas, “Differential inequalities for the positive zeros of Bessel functions,” J. Comput. Appl. Math. 30 (2), 139–143 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  25. E. K. Ifantis and P. D. Siafarikas, “A differential inequality for the positive zeros of Bessel functions,” J. Comput. Appl. Math. 44 (1), 115–120 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  26. M. E. H. Ismail, “The zeros of basic Bessel functions, the functions J? + ax(x) associated orthogonal polynomials,” J. Math. Anal. Appl. 86, 1–19 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  27. M. E. H. Ismail, “On zeros of Bessel functions,” Appl. Anal. 22, 167–168 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  28. M. E. H. Ismail and M. E. Muldoon, “On the variation with respect to a parameter of zeros of Bessel and q-Bessel functions,” J. Math. Anal. Appl. 135, 187–207 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  29. M. E. H. Ismail and M. E. Muldoon, “Bounds for the small real and purely imaginary zeros of Bessel and related functions,” Methods Appl. Anal. 2 (1), 1–21 (1995).

    MathSciNet  MATH  Google Scholar 

  30. F. H. Jackson, “The basic gamma function and elliptic function,” Proc. R. Soc. London Ser. A 76, 127–144 (1905).

    Article  MATH  Google Scholar 

  31. M. K. Kerimov, “The Rayleigh function: Theory and methods for its calculation,” Comput. Math. Math. Phys. 39 (12), 1883–1925 (1999).

    MathSciNet  MATH  Google Scholar 

  32. M. K. Kerimov, “Overview of some results concerning the theory and applications of the Rayleigh special function,” Comput. Math. Math. Phys. 48 (9), 1454–1507 (2008).

    Article  MathSciNet  Google Scholar 

  33. M. K. Kerimov, “Studies on the zeros of Bessel functions and methods for their computation,” Comput. Math. Math. Phys. 54 (9), 1337–1388 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  34. C. G. Kokologiannaki, M. E. Muldoon, and P. D. Siafarikas, “A unimodal property of purely imaginary zeros of Bessel and related functions,” Can. Math. Bull. 37 (3), 365–373 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  35. A. Laforgia and M. E. Muldoon, “Inequalities and approximations for zeros of Bessel functions of small order,” SIAM J. Math. Anal. 14 (2), 383–388 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Laforgia and M. E. Muldoon, “Monotonicity and concavity properties of zeros of Bessel functions,” J. Math. Anal. Appl. 98 (2), 470–477 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  37. J. T. Lewis and M. E. Muldoon, “Monotonicity and convexity properties of zeros of Bessel functions,” SIAM J. Math. Anal. 8 (1), 171–178 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  38. J. T. Lewis and J. V. Pule, “The free boson gas in a rotating bucket,” Comm. Math. Phys. 45, 115–131 (1975).

    Article  MathSciNet  Google Scholar 

  39. L. Lorch, “Elementary comparison techniques for certain classes of Sturm–Liouville equations,” Proceedings of International Conference on Differential Equations (Univ. Uppsaliensis, Uppsala, 1977), pp. 125–133.

    Google Scholar 

  40. L. Lorch, “Turanians and Wronskians for the zeros of Bessel functions,” SIAM J. Math. Anal. 11, 223–227 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  41. L. Lorch, M. E. Muldoon, and P. Szego, “Some monotonicity properties of Bessel functions,” SIAM J. Math. Anal. 4, 385–392 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  42. L. Lorch and P. Szego, “Higher monotonicity properties of certain Sturm–Liouville functions,” Acta Math. 109, 53–73 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  43. L. Lorch and P. Szego, “Higher monotonicity properties of certain Sturm–Liouville functions II,” Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 11, 455–457 (1963).

    MathSciNet  MATH  Google Scholar 

  44. L. Lorch and P. Szego, “On the zeros of derivatives of Bessel functions,” SIAM J. Math. Anal. 19 (6), 1450–1454 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  45. F. W. J. Olver (Ed.), Bessel Functions, Part 3: “Zeros and associated values” (Cambridge Univ. Press, Cambridge, 1960).

    MATH  Google Scholar 

  46. F. W. J. Olver, Asymptotics and Special Functions (Academic, New York, 1974; Nauka, Moscow, 1990).

    MATH  Google Scholar 

  47. R. Piessens, “A series expansion for the first positive zero of Bessel function,” Math. Comput. 42, 195–197 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  48. S. J. Putterman, M. Kac, and G. E. Uhlenbeck, “Possible origin of the quantized vortices in He II,” Phys. Rev. Lett. 29, 546–549 (1972).

    Article  Google Scholar 

  49. R. Ronveaux and A. Moussiaux, “A priori bound on the first zero of some special functions,” Ann. Soc. Sci. Bruxelles Ser. II 88, 169–175 (1974).

    MathSciNet  MATH  Google Scholar 

  50. J. A. Shohat and J. D. Tamarkin, The Problem of Moment, rev. edition (Am. Math. Soc., Providence, RI, 1950).

    MATH  Google Scholar 

  51. R. Spigler, “Alcuni risultati sugli zeri delle funzioni cilindriche e delle loro derivate,” Rend. Semin. Mat. Univ. Politech. Torino 38 (1), 67–85 (1980).

    MathSciNet  MATH  Google Scholar 

  52. F. G. Tricomi, “Sulle funzioni di Bessel di ordine e argomento pressoche uguali,” Atti Accad. Sci. Torino. Cl. Sci. Fis. Mat. Natur. 83, 3–20 (1949).

    MathSciNet  MATH  Google Scholar 

  53. J. C. F. Sturm, “Sur les équations differentielles linéares du second order,” J. Math. 1, 106–186 (1836).

    Google Scholar 

  54. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge Univ. Press, Cambridge, 1944).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control,”, Russian Academy of Sciences, Moscow, 119333, Russia

    M. K. Kerimov

Authors
  1. M. K. Kerimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. K. Kerimov.

Additional information

Original Russian Text © M.K. Kerimov, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 12, pp. 1986–2030.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kerimov, M.K. Studies on the zeros of Bessel functions and methods for their computation: 3. Some new works on monotonicity, convexity, and other properties. Comput. Math. and Math. Phys. 56, 1949–1991 (2016). https://doi.org/10.1134/S0965542516120125

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542516120125

Keywords

Search

Navigation