On integral functions with derivatives univalent in a circle
It is proved that if the increasing sequence $n_p$ of natural numbers satisfies the condition $n_{p+1}/n_p→1 (p→\infty)$ and all derivatives $f^{(n_p)}$ of the analytic function $f$ in $D=\{z : |z | < 1\}$ are univalent in $D$, then $f$ is an entire function. At the same time, for each...
Збережено в:
| Дата: | 1991 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1991
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/9622 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | It is proved that if the increasing sequence $n_p$ of natural numbers satisfies the condition $n_{p+1}/n_p→1 (p→\infty)$ and all derivatives $f^{(n_p)}$ of the analytic function $f$ in $D=\{z : |z | < 1\}$ are univalent in $D$, then $f$ is an entire function. At the same time, for each increasing sequence $(n_p)$ natural numbers such that $n_{p+1}/n_p→1 (p→\infty)$ there exists an analytic function $f$ in $D$ all of whose derivatives $f^{(n_p)}$ are univalent in $D$ and $\partial D$ is the boundary for $f$. The growth of entire functions with derivatives univalent in the disc $D$ is also studied. |
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