Abstract
Let bāCn\{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:tāC} with the unit ball Bn={zāC:|z|:=|z|12+ā¦+|zn|2<1} for any z0āBn. For this class of functions, there is introduced a concept of boundedness of L-index in the direction b, where L:BnāR+ is a positive continuous function such that L(z)>Ī²|b|1ā|z|, where Ī²>1 is some constant. For functions from this class, we describe a local behavior of modulus of directional derivatives on every ācircleā {z+tb:|t|=r/L(z)} with rā(0;Ī²],tāC,zāCn. 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:BnāR+.
Highlights
The theory of entire functions of bounded index was initiated by the paper of B
Let b ā Cn \ {0} be a given direction, L : Cn ā R+ be a continuous function. Is it possible to replace the condition āF is holomorphic in Cn ā by the condition āF is holomorphic on all slices z0 + tbā and to deduce all known properties of entire functions of bounded L-index in direction for this class of function class?
We prove several assertions that establish a connection between functions of bounded L-index in direction and functions of a bounded l-index of one variable
Summary
The theory of entire functions of bounded index was initiated by the paper of B. Let b ā Cn \ {0} be a given direction, L : Cn ā R+ be a continuous function Is it possible to replace the condition āF is holomorphic in Cn ā by the condition āF is holomorphic on all slices z0 + tbā and to deduce all known properties of entire functions of bounded L-index in direction for this class of function class?. Note that joint continuity and slice holomophy (in one direction b) do not imply holomorphy in a whole n-dimensional complex space (see examples in [13]) For these classes, the theory of a bounded index in the direction was constructed in papers [11,12,13]. All notations, introduced above for analytic functions of bounded L-index in the e b (Bn )
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