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Boundedness of L-Index for the Composition of Entire Functions of Several Variables

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Ukrainian Mathematical Journal Aims and scope

We consider the following compositions of entire functions

$$ F(z)=f\left(\varPhi (z)\right)\kern0.5em \mathrm{and}\kern0.5em H\left(z,w\right)=G\left({\varPhi}_1(z),{\varPhi}_1(w)\right), $$

where f : ℂ → ℂ, Φ : ℂn → ℂ, Φ1 : ℂn → ℂ, and Φ2 : ℂm → ℂ, and establish conditions guaranteeing the equivalence of boundedness of the l -index of a function f to the boundedness of the L-index of the function F in joint variables, where l : ℂ → ℝ+ is a continuous function and

$$ L(z)=\left(l\left(\varPhi (z)\right)|\frac{\partial \varPhi (z)}{\partial }|,\dots, l\left(\varPhi (z)\right)|\frac{\partial \varPhi (z)}{\partial }|\right). $$

Under certain additional restrictions imposed on the function H, we construct a function \( \tilde{L} \) such that H has a bounded \( \tilde{L} \)-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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Authors and Affiliations

  1. Ivano-Frankivs’k National Oil and Gas Technical University, Ivano-Frankivs’k, Ukraine

    A. I. Bandura

  2. Franko Lviv National University, Lviv, Ukraine

    O. B. Skaskiv

Authors
  1. A. I. Bandura
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  2. O. B. Skaskiv
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Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 10, pp. 1334–1344, October, 2018. , No. 2, 177–182 (2008).

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Bandura, A.I., Skaskiv, O.B. Boundedness of L-Index for the Composition of Entire Functions of Several Variables. Ukr Math J 70, 1538–1549 (2019). https://doi.org/10.1007/s11253-019-01589-9

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  • DOI: https://doi.org/10.1007/s11253-019-01589-9

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