Skip to main content
SpringerLink
Log in

On the Closest to Zero Roots and the Second Quotients of Taylor Coefficients of Entire Functions from the Laguerre–Pólya I Class

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

For an entire function \(f(z) = \sum _{k=0}^\infty a_k z^k, a_k>0\), we show that if f belongs to the Laguerre–Pólya class, and the quotients \(q_k:= \frac{a_{k-1}^2}{a_{k-2}a_k}, k=2, 3, \ldots \) satisfy the condition \(q_2 \le q_3\), then f has at least one zero in the segment \([-\frac{a_1}{a_2},0]\). We also give necessary conditions and sufficient conditions of the existence of such a zero in terms of the quotients \(q_k\) for \(k=2,3, 4\).

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. Bohdanov, A.: Determining bounds on the values of parameters for a function \(\varphi _{a}(z, m) =\sum _{k=0}^\infty \frac{z^k}{a^{k^2}}(k!)^m\), \(m \in (0,1), \) to belong to the Laguerre-Pólya class. Comput. Methods Funct. Theory (2017). https://doi.org/10.1007/s40315-017-0210-6

    Article  MATH  Google Scholar 

  2. Bohdanov, A., Vishnyakova, A.: On the conditions for entire functions related to the partial theta function to belong to the Laguerre–Pólya class. J. Math. Anal. Appl. 434(2), 1740–1752 (2016). https://doi.org/10.1016/j.jmaa.2015.09.084

    Article  MathSciNet  MATH  Google Scholar 

  3. Craven, T., Csordas, G.: Complex zero decreasing sequences. Methods Appl. Anal. 2, 420–441 (1995)

    MathSciNet  MATH  Google Scholar 

  4. Hirschman, I.I., Widder, D.V.: The Convolution Transform. Princeton University Press, Princeton (1955)

    MATH  Google Scholar 

  5. Hutchinson, J.I.: On a remarkable class of entire functions. Trans. Am. Math. Soc. 25, 325–332 (1923)

    Article  MathSciNet  Google Scholar 

  6. Katkova, O., Lobova, T., Vishnyakova, A.: On power series having sections with only real zeros. Comput. Methods Funct. Theory 3(2), 425–441 (2003)

    Article  MathSciNet  Google Scholar 

  7. Katkova, O., Lobova, T., Vishnyakova, A.: On entire functions having Taylor sections with only real zeros. J. Math. Phys. Anal. Geom. 11(4), 449–469 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Kostov, V.P.: About a partial theta function. C. R. Acad. Bulgare Sci. 66, 629–634 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Kostov, V.P.: On the zeros of a partial theta function. Bull. Sci. Math. 137(8), 1018–1030 (2013)

    Article  MathSciNet  Google Scholar 

  10. Kostov, V.P.: On the spectrum of a partial theta function. Proc. R. Soc. Edinb. Sect. A 144(05), 925–933 (2014)

    Article  MathSciNet  Google Scholar 

  11. Kostov, V.P.: Asymptotics of the spectrum of partial theta function. Revista Matematica Complutense 27(2), 677–684 (2014)

    Article  MathSciNet  Google Scholar 

  12. Kostov, V.P.: A property of a partial theta function. C. R. Acad. Bulgare Sci. 67, 1319–1326 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Kostov, V.P.: Asymptotic expansions of zeros of a partial theta function. C. R. Acad. Bulgare Sci. 68, 419–426 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Kostov, V.P.: On the double zeros of a partial theta function. Bulletin des Sciences Mathamatiques 140(4), 98–111 (2016)

    Article  MathSciNet  Google Scholar 

  15. Kostov, V.P.: On a partial theta function and its spectrum. Proc. R. Soc. Edinb. Sect. A Math. 146(3), 609–623 (2016)

    Article  MathSciNet  Google Scholar 

  16. Kostov, V.P.: The closest to 0 spectral number of the partial theta function. C. R. Acad. Bulgare Sci. 69, 1105–1112 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Kostov, V.P., Shapiro, B.: Hardy–Petrovitch–Hutchinson’s problem and partial theta function. Duke Math. J. 162(5), 825–861 (2013)

    Article  MathSciNet  Google Scholar 

  18. Levin, B.J.: Distribution of Zeros of Entire Functions, Transl. Math. Mono., 5. American Mathematical Society, Providence (1964); revised ed. (1980)

  19. Nguyen, T.H., Vishnyakova, A.: On the entire functions from the Laguerre–Pólya class having the decreasing second quotients of Taylor coefficients. J. Math. Anal. Appl. 465(1), 348–359 (2018)

    Article  MathSciNet  Google Scholar 

  20. Nguyen, T.H., Vishnyakova, A.: On the necessary condition for entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre-Pólya class. J. Math. Anal. Appl. (2019). https://doi.org/10.1016/j.jmaa.2019.123433

    Article  MATH  Google Scholar 

  21. Nguyen, T.H.: On the conditions for a special entire function relative to the partial theta-function and the Euler function to belong to the Laguerre–Pólya class. arXiv:1912.10035

  22. Obreschkov, N.: Verteilung und Berechnung der Nullstellen reeller Polynome. VEB Deutscher Verlag der Wissenschaften, Berlin (1963)

    Google Scholar 

  23. Ostrovskii, I.V.: On zero distribution of sections and tails of power series. Isr. Math. Conf. Proc. 15, 297–310 (2001)

    MathSciNet  MATH  Google Scholar 

  24. Pólya, G.: Collected Papers. In: Boas, R.P. (ed.) Location of Zeros, vol. II. MIT Press, Cambridge (1974)

    Google Scholar 

  25. Pólya, G., Schur, J.: Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144, 89–113 (1914)

    MathSciNet  MATH  Google Scholar 

  26. Pólya, G., Szegö, G.: Problems and Theorems in Analysis II. Springer Science and Business Media, Mathematics. Springer, Berlin (1997)

    Google Scholar 

  27. Warnaar, S.O.: Partial theta functions. https://www.researchgate.net/publication/327791878_Partial_theta_functions. Accessed 22 May 2020

Download references

Acknowledgements

The first author is deeply grateful to the Akhiezer Foundation for the 2019 financial support.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine

    Thu Hien Nguyen & Anna Vishnyakova

Authors
  1. Thu Hien Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Vishnyakova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thu Hien Nguyen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

T. H. Nguyen: Partially supported by the Akhiezer Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nguyen, T.H., Vishnyakova, A. On the Closest to Zero Roots and the Second Quotients of Taylor Coefficients of Entire Functions from the Laguerre–Pólya I Class. Results Math 75, 115 (2020). https://doi.org/10.1007/s00025-020-01245-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-020-01245-w

Keywords

Mathematics Subject Classification

Search

Navigation