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Journal of Mathematical Sciences

, Volume 244, Issue 1, pp 1–21 | Cite as

Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables

  • Andriy BanduraEmail author
  • Oleh Skaskiv
Article
  • 1 Downloads

Abstract

We investigate the slice holomorphic functions of several complex variables that have a bounded L-index in some direction and are entire on every slice {z0 + tb : t ∈ ℂ} for every z0 ∈ ℂn and for a given direction b ∈ ℂn\ {0}. For this class of functions, we prove some criteria of boundedness of the L-index in direction describing a local behavior of the maximum and minimum moduli of a slice holomorphic function and give estimates of the logarithmic derivative and the distribution of zeros. Moreover, we obtain analogs of the known Hayman theorem and logarithmic criteria. They are applicable to the analytic theory of differential equations. We also study the value distribution and prove the existence theorem for those functions. It is shown that the bounded multiplicity of zeros for a slice holomorphic function F : ℂn → ℂ is the necessary and sufficient condition for the existence of a positive continuous function L : ℂn → ℝ+ such that F has a bounded L-index in direction.

Keywords

Bounded index bounded L-index in direction slice function holomorphic function maximum modulus minimum modulus bounded l-index existence theorem distribution of zeros logarithmic derivative directional derivative 

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Ivano-Frankivs’k National Technical University of Oil and GasIvano-Frankivs’kUkraine
  2. 2.Ivan Franko National University of LvivLvivUkraine

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