The minimal growth of entire functions with given zeros along unbounded sets

  • I. V. Andrusyak Lviv Polytechnic National University, Lviv, Ukraine
  • P.V. Filevych Lviv Politechnic National University, Lviv
Keywords: entire function; maximum modulus; zeros; counting function

Abstract

Let $l$ be a continuous function on $\mathbb{R}$ increasing to $+\infty$, and $\varphi$ be a positive function on $\mathbb{R}$. We proved that the condition
$$
\varliminf_{x\to+\infty}\frac{\varphi(\ln[x])}{\ln x}>0
$$
is necessary and sufficient in order that for any complex sequence $(\zeta_n)$ with $n(r)\ge l(r)$, $r\ge r_0$, and every set $E\subset\mathbb{R}$ which is unbounded from above there exists an entire function $f$ having zeros only at the points $\zeta_n$ such that
$$
\varliminf_{r\in E,\ r\to+\infty}\frac{\ln\ln M_f(r)}{\varphi(\ln n_\zeta(r))\ln l^{-1}(n_\zeta(r))}=0.
$$
Here $n(r)$ is the counting function of $(\zeta_n)$, and $M_f(r)$ is the maximum modulus of $f$.

References

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Published
2020-12-25
How to Cite
1.
Andrusyak IV, Filevych P. The minimal growth of entire functions with given zeros along unbounded sets. Mat. Stud. [Internet]. 2020Dec.25 [cited 2021Oct.16];54(2):146-53. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/160
Section
Articles