The growth of entire functions with zero sets having integer-valued exponent of convergence (in Ukrainian)

I.V.Andrusyak; P.V.Filevych

Let $\zeta=(z_n)$ be a sequence of complex numbers tending to $\infty$, $n_\zeta(r)$ be its counting function, $\tau_\zeta$ be its exponent of convergence, $A(\zeta)$ be the class of entire functions whose zero sets coincide with $\zeta,$ be a function that is positive continuous and increasing to ${+\infty}$ on ${\mathbb R}$, and $\tau\in{\mathbb N}$. In particular, we prove the next assertion: for any sequence $\zeta$ with $\tau_\zeta=\tau$ there exists an entire function $f\in A(\zeta)$ such that $\displaystyle \varliminf_{r\to{+\infty}}\frac{\ln M_f(r)}{n_\zeta(r) \varphi(\ln n_\zeta(r))}=0 $ if and only if $\int_0^\infty{dx}/{\varphi(x)}<\infty$.

1. Гольдберг А. А., Островский И. В. Распределение значений мероморфных функций.~- М.: Наука, 1970. - 592 с.

2. Bergweiler W. Canonical products of infinite order// J. reine angew. Math. - 1992. - V. 430. - P. 85-107.

3. Miles J. On the growth of entire functions with zero sets having infinite exponent of convergence// Ann. Acad. Sci. Fenn. Math. - 2002. - V. 27. - P. 69-90.

4. Хейман У. Мероморфные функции. - М.: Мир, 1966. - 288 с.

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

6. Sheremeta M. M. A remark to the construction of canonical products of minimal growth// Mat. fizika, analiz, geometriya. - 2004. - V. 11, № 2 - P. 243-248.

7. Хабибуллин Б. Н. Рост целых функций с заданными нулями и представление мероморфных функций// Мат. заметки. - 2003. - Т. 73. - P. 120-134.

	The growth of entire functions with zero sets having integer-valued exponent of convergence (in Ukrainian)
Author	I.V.Andrusyak, P.V.Filevych Lviv Polytechnic National University, Stepan Gzhytskyi National University of Veterinary Medicine and Biotechnologies of Lviv, Ivan Franko National University of Lviv
Abstract	Let $\zeta=(z_n)$ be a sequence of complex numbers tending to $\infty$, $n_\zeta(r)$ be its counting function, $\tau_\zeta$ be its exponent of convergence, $A(\zeta)$ be the class of entire functions whose zero sets coincide with $\zeta,$ be a function that is positive continuous and increasing to ${+\infty}$ on ${\mathbb R}$, and $\tau\in{\mathbb N}$. In particular, we prove the next assertion: for any sequence $\zeta$ with $\tau_\zeta=\tau$ there exists an entire function $f\in A(\zeta)$ such that $\displaystyle \varliminf_{r\to{+\infty}}\frac{\ln M_f(r)}{n_\zeta(r) \varphi(\ln n_\zeta(r))}=0 $ if and only if $\int_0^\infty{dx}/{\varphi(x)}<\infty$.
Keywords	entire function, growth, convergence exponent, zero set
DOI	doi:10.30970/ms.32.1.12-20
Reference	1. Гольдберг А. А., Островский И. В. Распределение значений мероморфных функций.~- М.: Наука, 1970. - 592 с. 2. Bergweiler W. Canonical products of infinite order// J. reine angew. Math. - 1992. - V. 430. - P. 85-107. 3. Miles J. On the growth of entire functions with zero sets having infinite exponent of convergence// Ann. Acad. Sci. Fenn. Math. - 2002. - V. 27. - P. 69-90. 4. Хейман У. Мероморфные функции. - М.: Мир, 1966. - 288 с. 5. Andrusyak I. V., Filevych P. V. The growth of an entire function with a given sequence of zeros// Mat. Stud. - 2008. - V. 30, № 2 - P. 115-124. 6. Sheremeta M. M. A remark to the construction of canonical products of minimal growth// Mat. fizika, analiz, geometriya. - 2004. - V. 11, № 2 - P. 243-248. 7. Хабибуллин Б. Н. Рост целых функций с заданными нулями и представление мероморфных функций// Мат. заметки. - 2003. - Т. 73. - P. 120-134.
Pages	12-20
Volume	32
Issue	1
Year	2009
Journal	Matematychni Studii
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