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

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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.

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Authors and Affiliations

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

    Andriy Bandura

  2. Ivan Franko National University of Lviv, Lviv, Ukraine

    Oleh Skaskiv

Authors
  1. Andriy Bandura
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  2. Oleh Skaskiv
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Correspondence to Andriy Bandura.

Additional information

The paper is dedicated to the 100th anniversary of G. D. Suvorov

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 16, No. 2, pp. 154–180 April–June, 2019.

The authors are thankful to Professor S. Yu. Favorov (Kharkiv) for the formulation of the attractive problem.

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Bandura, A., Skaskiv, O. Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables. J Math Sci 244, 1–21 (2020). https://doi.org/10.1007/s10958-019-04600-7

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  • DOI: https://doi.org/10.1007/s10958-019-04600-7

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