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Open AccessArticle

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.
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 bCn\{0} 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 {z0+tb:tC} with the unit ball Bn={zC:|z|:=|z|12++|zn|2<1} for any z0Bn. For this class of functions, there is introduced a concept of boundedness of L-index in the direction b, where L:BnR+ is a positive continuous function such that L(z)>β|b|1|z|, where β>1 is some constant. For functions from this class, we describe a local behavior of modulus of directional derivatives on every ’circle’ {z+tb:|t|=r/L(z)} with r(0;β],tC,zCn. 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 b of the slice holomorphic function F and the boundedness of lz-index of the slice function gz(t)=F(z+tb) with lz(t)=L(z+tb). 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 L:BnR+. View Full-Text
Keywords: bounded index; bounded L-index in direction; slice function; analytic function; bounded l-index; unit ball; local behavior; maximum modulus bounded index; bounded L-index in direction; slice function; analytic function; bounded l-index; unit ball; local behavior; maximum modulus
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 Style

Bandura, 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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