Pflugertype theorem for functions of refined regular growth 

Author 
chyzhykov@yahoo.com
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
$\logf(re^{i\theta})=r^\rho h_f(\theta)+ O(\frac{r^{\rho}}{\delta(r)})$, $ 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.47.2.169178

Reference 
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Pages 
169178

Volume 
47

Issue 
2

Year 
2017

Journal 
Matematychni Studii

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