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# Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction

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Department of Advanced Mathematics, Ivano-Frankivsk National Technical University of Oil and Gas, 76019 Ivano-Frankivsk, Ukraine
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Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine
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Author to whom correspondence should be addressed.
Axioms 2019, 8(3), 88; https://doi.org/10.3390/axioms8030088
Received: 25 June 2019 / Revised: 24 July 2019 / Accepted: 25 July 2019 / Published: 26 July 2019
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
In this paper, for a given direction $b ∈ C n \ { 0 }$ we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line ${ z 0 + t b : t ∈ C }$ for any $z 0 ∈ C n$ . Unlike to quaternionic analysis, we fix the direction $b$ . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable $z 1$ and continuous in variable $z 2 .$ For this class of functions there is introduced a concept of boundedness of L-index in the direction $b$ where $L : C n → R +$ is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function $L : C n → R + .$ View Full-Text
MDPI and ACS Style

Bandura, A.; Skaskiv, O. Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction. Axioms 2019, 8, 88. https://doi.org/10.3390/axioms8030088

AMA Style

Bandura A, Skaskiv O. Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction. Axioms. 2019; 8(3):88. https://doi.org/10.3390/axioms8030088

Chicago/Turabian Style

Bandura, Andriy, and Oleh Skaskiv. 2019. "Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction" Axioms 8, no. 3: 88. https://doi.org/10.3390/axioms8030088

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