Boundedness of $L$-index for the composition of entire functions of several variables
We consider the following compositions of entire functions $F(z) = f \bigl( \Phi (z)\bigr) $ and $H(z,w) = G(\Phi 1(z),\Phi 2(w))$, where f$f : C \rightarrow C, \Phi : C^n \rightarrow C,\; \Phi_1 : C^n \rightarrow C, \Phi_2 : C^m \rightarrow C$, and establish conditions guaranteeing the equival...
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| Date: | 2018 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2018
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1639 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | We consider the following compositions of entire functions $F(z) = f \bigl( \Phi (z)\bigr) $
and $H(z,w) = G(\Phi 1(z),\Phi 2(w))$, where
f$f : C \rightarrow C, \Phi :
C^n \rightarrow C,\; \Phi_1 : C^n \rightarrow C, \Phi_2 : C^m \rightarrow C$, and establish conditions guaranteeing the equivalence of boundedness of the $l$-index of the function $f$ to the boundedness of the $L$-index of the function $F$ in joint variables, where $l$ :
$C \rightarrow R_{+}$ is a continuous function and $$L(z) = \Bigl( l\bigl( \Phi (z)\bigr)
\bigm| \frac{\partial \Phi (z)}{\partial z_1}\bigm| ,..., l \bigl( \Phi (z) \bigr) \bigm|\frac{\partial \Phi (z)}{\partial z_n} \bigm| \Bigr).$$
Under certain additional
restrictions imposed on the function $H$, we construct a function $\widetilde{L} $ such that $H$ has a bounded $\widetilde{ L}$ -index in joint variables
provided that the function $G$ has a bounded $L$-index in joint variables. This solves a problem posed by Sheremeta. |
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