Minimal growth of entire functions with prescribed zeros outside exceptional sets

Author:

Andrusyak I.,Filevych P.,O. Oryshchyn

Abstract

Let $h$ be a positive continuous increasing to $+\infty$ function on $\mathbb{R}$. It is proved that for an arbitrary complex sequence $(\zeta_n)$ such that $0<|\zeta_1|\le|\zeta_2|\le\dots$ and $\zeta_n\to\infty$ as $n\to\infty$, there exists an entire function $f$ whose zeros are the $\zeta_n$, with multiplicities taken into account, for which$$\ln m_2(r,f)=o(N(r)),\quad r\notin E,\ r\to+\infty.$$with a set $E$ satisfying $\int_{E\cap(1,+\infty)}h(r)dr<+\infty$, if and only if $\ln h(r)=O(\ln r)$ as $r\to+\infty$.Here $N(r)$ is the integrated counting function of the sequence $(\zeta_n)$ and$$m_2(r,f)=\left(\frac{1}{2\pi}\int_0^{2\pi}|\ln|f(re^{i\theta})||^2d\theta\right)^{1/2}.$$

Publisher

Ivan Franko National University of Lviv

Subject

General Mathematics

Reference21 articles.

1. A.A. Gol’dberg, The representation of a meromorphic function in the form of a quotient of entire functions, Izv. Vyssh. Uchebn. Zaved. Mat., 10 (1972), 13–17. (in Russian)

2. I.V. Andrusyak, P.V. Filevych, The growth of an entire function with a given sequence of zeros, Mat. Stud., 30 (2008), №2, 115–124.

3. W. Bergweiler, Canonical products of infinite order, J. Reine Angew. Math., 430 (1992), 85–107. doi.org/10.1515/crll.1992.430.85

4. W. Bergweiler, A question of Gol’dberg concerning entire functions with prescribed zeros, J. Anal. Math., 63 (1994), №1, 121–129. doi.org/10.1007/BF03008421

5. J. Miles, On the growth of entire functions with zero sets having infinite exponent of convergence, Ann. Acad. Sci. Fenn. Math., 27 (2002), 69–90.

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