TY - JOUR
AU - Andrusyak, I.V.
AU - Filevych, P.V.
PY - 2024/02/26
Y2 - 2024/08/06
TI - Comparative growth of an entire function and the integrated counting function of its zeros
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 16
IS - 1
SE - Scientific articles
DO - 10.15330/cmp.16.1.5-15
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/6976
SP - 5-15
AB - <p>Let $(\zeta_n)$ be a sequence of complex numbers such that $\zeta_n\to\infty$ as $n\to\infty$, $N(r)$ be the integrated counting function of this sequence, and let $\alpha$ be a positive continuous and increasing to $+\infty$ function on $\mathbb{R}$ for which $\alpha(r)=o(\log (N(r)/\log r))$ as $r\to+\infty$. It is proved that for any set $E\subset(1,+\infty)$ satisfying $\int_{E}r^{\alpha(r)}dr=+\infty$, there exists an entire function $f$ whose zeros are precisely the $\zeta_n$, with multiplicities taken into account, such that the relation $$ \liminf_{r\in E,\ r\to+\infty}\frac{\log\log M(r)}{\log r\log (N(r)/\log r)}=0 $$ holds, where $M(r)$ is the maximum modulus of the function $f$. It is also shown that this relation is best possible in a certain sense.</p>
ER -