Slice Holomorphic Functions in the Unit Ball Having a Bounded L-Index in Direction
1
Department of Advanced Mathematics, Ivano-Frankivsk National Technical University of Oil and Gas, 76019 Ivano-Frankivsk, Ukraine
2
Faculty of Mathematics and Computer Sciences, Vasyl Stefanyk Precarpathian National University, 76018 Ivano-Frankivsk, Ukraine
3
Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine
*
Author to whom correspondence should be addressed.
Axioms 2021, 10(1), 4; https://doi.org/10.3390/axioms10010004
Received: 26 November 2020 / Revised: 22 December 2020 / Accepted: 25 December 2020 / Published: 30 December 2020
(This article belongs to the Collection Mathematical Analysis and Applications)
Let be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e., we study functions that are analytic in the intersection of every slice with the unit ball for any . For this class of functions, there is introduced a concept of boundedness of L-index in the direction , where is a positive continuous function such that where is some constant. For functions from this class, we describe a local behavior of modulus of directional derivatives on every ’circle’ with . It is estimated by the value of the function at the center of the circle. Other propositions concern a connection between the boundedness of L-index in the direction of the slice holomorphic function F and the boundedness of -index of the slice function with In addition, we show that every slice holomorphic and joint continuous function in the unit ball has a bounded L-index in direction in any domain compactly embedded in the unit ball and for any continuous function
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Keywords:
bounded index; bounded L-index in direction; slice function; analytic function; bounded l-index; unit ball; local behavior; maximum modulus
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MDPI and ACS Style
Bandura, A.; Martsinkiv, M.; Skaskiv, O. Slice Holomorphic Functions in the Unit Ball Having a Bounded L-Index in Direction. Axioms 2021, 10, 4. https://doi.org/10.3390/axioms10010004
AMA Style
Bandura A, Martsinkiv M, Skaskiv O. Slice Holomorphic Functions in the Unit Ball Having a Bounded L-Index in Direction. Axioms. 2021; 10(1):4. https://doi.org/10.3390/axioms10010004
Chicago/Turabian StyleBandura, Andriy; Martsinkiv, Maria; Skaskiv, Oleh. 2021. "Slice Holomorphic Functions in the Unit Ball Having a Bounded L-Index in Direction" Axioms 10, no. 1: 4. https://doi.org/10.3390/axioms10010004
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