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Ufa Mathematical Journal, 2021, Volume 13, Issue 1, Pages 31–45
DOI: https://doi.org/10.13108/2021-13-1-31
(Mi ufa549)
 

This article is cited in 4 scientific papers (total in 4 papers)

Joint estimates for zeros and Taylor coefficients of entire function

G. G. Braichev

Moscow State Pedagogical University, Malaya Pirogovskaya str. 1, bld. 1, 119991, Moscow, Russia
References:
Abstract: In the paper, for an entire function $f(z)=\sum\limits_{n=0}^{\infty} f_n z^n$, we provide asymptotic and uniform bounds of commensurability of the growth of zeroes and the decaying of the Taylor coefficients one with respect to the other. As an initial point for these studies, the following Hadamard statement serves: if the coefficients of the series obey the inequality $|f_n|\leqslant\varphi(n)$ with some function $\varphi(x),$ then the absolute values of the zeroes grows faster than $1/\sqrt[n]{\varphi(n)}.$ In the present work we improve recently obtained lower bound for the joint growth of the zeroes and the coefficients via the maximal term of the Taylor series of the function $f(z)$ or via the counting function of its zeroes. The employing of Hadamard-rectified coefficients of the series give an opportunity to establish corresponding two-sided estimates. By the methods developing classical ideas we find a numerical dependence of such estimates on the sizes of the gaps of the power series representing the entire function. In particular, we find asymptotic identities relating the zeroes and the coefficients of an entire function. The obtained estimates are sharp and strengthen the known results by other authors.
Keywords: Taylor coefficients, Hadamard-rectified zeroes of entire function.
Funding agency Grant number
Russian Foundation for Basic Research 18-01-00236
The reported study was funded by RFBR according to the research project no. 18-01-00236.
Received: 15.11.2020
Russian version:
Ufimskii Matematicheskii Zhurnal, 2021, Volume 13, Issue 1, Pages 31–45
Bibliographic databases:
Document Type: Article
UDC: 517.537.3
MSC: 30D20
Language: English
Original paper language: Russian
Citation: G. G. Braichev, “Joint estimates for zeros and Taylor coefficients of entire function”, Ufimsk. Mat. Zh., 13:1 (2021), 31–45; Ufa Math. J., 13:1 (2021), 31–45
Citation in format AMSBIB
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\by G.~G.~Braichev
\paper Joint estimates for zeros and Taylor coefficients of entire function
\jour Ufimsk. Mat. Zh.
\yr 2021
\vol 13
\issue 1
\pages 31--45
\mathnet{http://mi.mathnet.ru/ufa549}
\transl
\jour Ufa Math. J.
\yr 2021
\vol 13
\issue 1
\pages 31--45
\crossref{https://doi.org/10.13108/2021-13-1-31}
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Linking options:
  • https://www.mathnet.ru/eng/ufa549
  • https://doi.org/10.13108/2021-13-1-31
  • https://www.mathnet.ru/eng/ufa/v13/i1/p31
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Уфимский математический журнал
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    English version PDF:19
    References:33
     
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