# Pfluger-type theorem for functions of refined regular growth

Author
Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract
Without a priori assumptions on zero distribution we prove that if an entire function $f$ of noninteger order $\rho$ has an asymptotic of the form $\log|f(re^{i\theta})|=r^\rho h_f(\theta)+ O(\frac{r^{\rho}}{\delta(r)})$, $E\not \ni re^{i\theta}\to \infty$, where $h$ is the indicator of $f$, $\delta$ is an unbounded regularly growing function, and $E$ is an appropriate exceptional set, then the counting function of zeros and the integrated counting function of zeros in the angle $\{z: \alpha\le\arg z\le\beta\}$ have similar asymptotic for almost all $\alpha\le\beta$. It complements results on functions of completely regular growth due to P. Agranovich and V. Logvinenko, B. Vynnyts'kyi and R. Khats'.
Keywords
entire function; completely regular growth; indicator; refined regular growth; zero distribution
DOI
doi:10.15330/ms.47.2.169-178
Reference
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Pages
169-178
Volume
47
Issue
2
Year
2017
Journal
Matematychni Studii
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