Skip to main content
SpringerLink
Log in

On the Connection between the Growth of Zeros and the Decrease of Taylor Coefficients of Entire Functions

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

In the theory of growth of entire functions, two trends have historically developed. The first trend deals with the calculation or estimation of the growth characteristics of the maximum modulus of an entire function (order, type, etc.) in terms of its Taylor series coefficients. In the second trend, the dependence of the growth of a function on the distribution of its zeros is studied. The aim of the present paper is to consider direct connections between the zeros and Taylor coefficients of an entire function, considering both classical and recent advances in the topic.

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

Notes

  1. Note that the equality \(\varliminf_{n\to\infty} \sqrt[n]{|f_n|\varphi_n}= \varliminf_{n\to\infty}\sqrt[n]{\widehat{f}_{n}\varphi_n}(>0)\) does not hold in the general case, because the sequence \(\Phi=\{f_n\}_{n\in\mathbb{N}_0}\) may contain a subsequence consisting of zeros.

  2. A value \(a\in\mathbb{C}\) is said to be Borel exceptional for an entire function \(f\) if the category of growth of the counting function of the set of its \(a\)-points is lower than the category of growth of \(\ln M_f(r)\) (see [4, pp. 62, 69 (Russian version)]).

  3. At multiple points, the values of the corresponding derivatives also coincide.

  4. It suffices to consider the difference of these functions.

  5. The asymptotic behavior of zeros of a lacunar Taylor series of special type was studied by other methods in Hardy’s paper [10].

  6. \(\rho\)-densities are \(\rho\) times larger than the previously introduced densities for \(h(r)=r^\rho\), which is easy to see from condition (1.3).

References

  1. G. Valiron, “Sur les fonctions entières d’ordre nul et d’ordre fini et en particulier les fonctions à correspondance règulière,” Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3) 5, 117–257 (1913).

    MathSciNet  MATH  Google Scholar 

  2. G. G. Braichev, “Joint estimates for zeros and Taylor coefficients of entire function,” Ufa Math. J. 13 (1), 31–45 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  3. G. G. Braichev, Extremal Problems in the Theory of the Relative Growth of Convex and Entire Functions, Doctoral (Phys.–Math.) Dissertation (RUDN, Moscow, 2018) [in Russian].

    Google Scholar 

  4. A. A. Gol’dberg and I. V. Ostrovskii, The Distribution of Values of Meromorphic Functions (Nauka, Moscow, 1970) [in Russian].

    Google Scholar 

  5. A. Yu. Popov, “Development of the Valiron–Levin theorem on the least possible type of entire functions with a given upper \(\rho\)-density of roots,” Journal of Mathematical Sciences 211 (4), 579–616 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  6. S. M. Shach, “On integral functions of perfectly regular growth,” J. London Math. Soc. 14, 293–302 (1939).

    Article  MathSciNet  Google Scholar 

  7. B. Ya. Levin, Distribution of Zeros of Entire Functions (GITTL, Moscow, 1956) [in Russian].

    MATH  Google Scholar 

  8. B. N. Khabibullin, “Zero sequences of holomorphic functions, representation of meromorphic functions. II. Entire functions,” Sb. Math. 200 (2), 283–312 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  9. G. G. Braichev, “The exact bounds of lower type magnitude for entire function of order \(\rho\in(0,1)\) with zeros of prescribed average densities,” Ufa Math. J. 7 (4), 32–57 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  10. G. H. Hardy, “On the zeroes of certain classes of integral Taylor series. I,” Proc. London Math. Soc. 2 (1), 332–339 (1905).

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Yu. Popov, “The least possible type under the order \(\rho<1\) of canonical products with positive zeros of a given upper \(\rho\)-density,” Moscow Univ. Math. Bull. 60 (1), 32–36 (2005).

    MathSciNet  Google Scholar 

  12. G. G. Braichev and V. B. Sherstyukov, “On the least possible type of entire functions of order \(\rho\in(0,1)\) with positive zeros,” Izv. Math. 75 (1), 1–27 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  13. G. G. Braichev, “The least type of an entire function of order \(\rho\in(0,1)\) having positive zeros with prescribed averaged densities,” Sb. Math. 203 (7), 950–975 (2012).

    Article  MathSciNet  Google Scholar 

  14. G. G. Braichev and V. B. Sherstyukov, “Sharp bounds for asymptotic characteristics of growth of entire functions with zeros on given sets,” Fundam. Prikl. Mat. 22 (1), 51–97 (2018).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Pedagogical State University, Moscow, 119882, Russia

    G. G. Braichev

Authors
  1. G. G. Braichev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. G. Braichev.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 32–45 https://doi.org/10.4213/mzm13559.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Braichev, G.G. On the Connection between the Growth of Zeros and the Decrease of Taylor Coefficients of Entire Functions. Math Notes 113, 27–38 (2023). https://doi.org/10.1134/S0001434623010042

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434623010042

Keywords

Search

Navigation