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Matematicheskie Zametki, 1973, Volume 14, Issue 2, Pages 173–184 (Mi mzm7246)  

Asymptote of some entire exponential-type functions with zeros on spirals

S. K. Balashov

Rostov State University
Abstract: The author considers the Weierstrass canonical product of the first kind $\Pi(z)$, all roots of which lie on a spiral with equation in polar coordinates $(r,\Phi):\Phi=\ln\ln r$. With certain additional conditions on the roots, the asymptote is found for the function $\ln\{e^{Az}\Pi(z)\}$ ($A$ is some constant) in the complex plane cut along the spiral $\Phi=\ln\ln r$. The result is applied to the question of the sufficient condition for the satisfaction of an inequality for exponential-type functions, used in questions of the Dirichlet-series representation of analytic functions.
Received: 15.11.1971
English version:
Mathematical Notes, 1973, Volume 14, Issue 2, Pages 658–664
DOI: https://doi.org/10.1007/BF01147109
Bibliographic databases:
UDC: 517.5
Language: Russian
Citation: S. K. Balashov, “Asymptote of some entire exponential-type functions with zeros on spirals”, Mat. Zametki, 14:2 (1973), 173–184; Math. Notes, 14:2 (1973), 658–664
Citation in format AMSBIB
\Bibitem{Bal73}
\by S.~K.~Balashov
\paper Asymptote of some entire exponential-type functions with zeros on spirals
\jour Mat. Zametki
\yr 1973
\vol 14
\issue 2
\pages 173--184
\mathnet{http://mi.mathnet.ru/mzm7246}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=333171}
\zmath{https://zbmath.org/?q=an:0279.30027}
\transl
\jour Math. Notes
\yr 1973
\vol 14
\issue 2
\pages 658--664
\crossref{https://doi.org/10.1007/BF01147109}
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    Математические заметки Mathematical Notes
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