On the Radii of univalence of Gel'fond-Leont'ev derivatives
Let $0 < R < +\infty,$ let $A(R)$ bethe class of functions $$f(z) = \sum_{k=0}^{\infty}f_kz^k,$$ analytic in $\{ z: |z| < R \}$, and let $$l(z) = \sum_{k=0}^{\infty}l_kz^k,\; l_k > 0$$ be a formal power series. We prove that if $l^2_k/l_{k+1}l_{k-1}$ is a nonincreasing s...
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| Datum: | 1995 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1995
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/5428 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Institution
Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | Let $0 < R < +\infty,$ let $A(R)$ bethe class of functions
$$f(z) = \sum_{k=0}^{\infty}f_kz^k,$$
analytic in $\{ z: |z| < R \}$, and let
$$l(z) = \sum_{k=0}^{\infty}l_kz^k,\; l_k > 0$$
be a formal power series.
We prove that if $l^2_k/l_{k+1}l_{k-1}$ is a nonincreasing sequence, $f \in A(R)$, and
$|f_k/f_{k+1} \nearrow R,\; k \rightarrow \infty,\; 0 < R < +\infty$, then the sequence $(\rho_n)$ of radii of univalence of the Gel'fondLeont'ev derivatives
satisfies the relation
$$D^n_lf(z) = \sum_{k=0}^{\infty}\frac{l_kf_{k+n}}{l_{k+n}}z_k$$
The case where the condition $|f_k/f_{k+1}|\nearrow R,\quad k \rightarrow \infty$, is not satisfied is also considered. |
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