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

  • A. I. Bandura
  • O. B. Skaskiv
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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 :  → , Φ : n → , Φ1 : n → , and Φ2 : m → 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 :  → + 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

  • A. I. Bandura
    • 1
  • O. B. Skaskiv
    • 2
  1. 1.Ivano-Frankivs’k National Oil and Gas Technical UniversityIvano-Frankivs’kUkraine
  2. 2.Franko Lviv National UniversityLvivUkraine

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