Working Paper Article Version 1 This version is not peer-reviewed

# Slice Holomorphic Functions in the Unit Ball Having Bounded L-Index in Direction

Version 1 : Received: 26 November 2020 / Approved: 27 November 2020 / Online: 27 November 2020 (20:16:41 CET)

A peer-reviewed article of this Preprint also exists.

Bandura, A.; Martsinkiv, M.; Skaskiv, O. Slice Holomorphic Functions in the Unit Ball Having a Bounded L-Index in Direction. Axioms 2021, 10, 4. Bandura, A.; Martsinkiv, M.; Skaskiv, O. Slice Holomorphic Functions in the Unit Ball Having a Bounded L-Index in Direction. Axioms 2021, 10, 4.

## Abstract

Let b∈Cn∖{0} be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice {z0+tb:t∈C} with the unit ball Bn={z∈C:|z|:=|z|12+…+|zn|2&lt;1} for any z0∈Bn. For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L:Bn→R+ is a positive continuous function such that L(z)&gt;β|b|1−|z|, where β&gt;1 is some constant. For functions from this class we describe local behavior of modulus of directional derivatives on every ’circle’ {z+tb:|t|=r/L(z)} with r∈(0;β],t∈C,z∈Cn. It is estimated by value of the function at center of the circle. Other propositions concern a connection between boundedness of L-index in the direction b of the slice holomorphic function F and boundedness of lz-index of the slice function gz(t)=F(z+tb) with lz(t)=L(z+tb). Also we show that every slice holomorphic and joint continuous function in the unit ball has bounded L-index in direction in any domain compactly embedded in the unit ball and for any continuous function L:Bn→R+.

## Keywords

bounded index; bounded L-index in direction; slice function; analytic function; bounded l-index; unit ball; local behavior; maximum modulus

## Subject

Computer Science and Mathematics, Algebra and Number Theory