Asymptotics of the entire functions with $\upsilon$-density of zeros along the logarithmic spirals

Authors

  • M.V. Zabolotskyj Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
  • Yu.V. Basiuk Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine https://orcid.org/0000-0002-6141-8975
https://doi.org/10.15330/cmp.11.1.26-32

Keywords:

entire function, density of zeros, logarithmic spiral
Published online: 2019-06-30

Abstract

Let $\upsilon$ be the growth function such that $r\upsilon'(r)/\upsilon (r) \to 0$ as $r \to +\infty$, $l_\varphi^c = \{z=te^{i(\varphi+c \ln t)}, 1 \leqslant t < +\infty\}$ be the logarithmic spiral, $f$ be the entire function of zero order. The asymptotics of $\ln f(re^{i(\theta +c \ln r)})$ along ordinary logarithmic spirals $l_\theta^c$ of the function $f$ with $\upsilon$-density of zeros along $l_\varphi^c$ outside the $C_0$-set is found. The inverse statement is true just in case zeros of $f$ are placed on the finite logarithmic spirals system $\Gamma_m = \bigcup_{j=0}^m l_{\theta_j}^c$.

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How to Cite
(1)
Zabolotskyj, M.; Basiuk, Y. Asymptotics of the Entire Functions With $\upsilon$-Density of Zeros Along the Logarithmic Spirals. Carpathian Math. Publ. 2019, 11, 26-32.