Slice Holomorphic Functions in Several Variables with Bounded <i>L</i>-Index in Direction

• Main Authors: Andriy Bandura, Oleh Skaskiv
• Format: Article
• Language: English
• Published: MDPI AG 2019-07-01
• Series: Axioms
• Subjects: bounded index; bounded <i>L</i>-index in direction; slice function; entire function; bounded <i>l</i>-index
• Online Access: https://www.mdpi.com/2075-1680/8/3/88

In this paper, for a given direction <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="bold">b</mi> <mo>&#8712;</mo> <msup> <mi mathvariant="double-struck">C</mi> <mi>n</mi> </msup> <mo>\</mo> <mrow> <mo>{</mo> <mn mathvariant="bold">0</mn> <mo>}</mo> </mrow> </mrow> </semantics> </math> </inline-formula> we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line <inline-formula> <math display="inline"> <semantics> <mrow> <mo>{</mo> <msup> <mi>z</mi> <mn>0</mn> </msup> <mo>+</mo> <mi>t</mi> <mi mathvariant="bold">b</mi> <mo>:</mo> <mi>t</mi> <mo>&#8712;</mo> <mi mathvariant="double-struck">C</mi> <mo>}</mo> </mrow> </semantics> </math> </inline-formula> for any <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>z</mi> <mn>0</mn> </msup> <mo>&#8712;</mo> <msup> <mi mathvariant="double-struck">C</mi> <mi>n</mi> </msup> </mrow> </semantics> </math> </inline-formula>. Unlike to quaternionic analysis, we fix the direction <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">b</mi> </semantics> </math> </inline-formula>. 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 <inline-formula> <math display="inline"> <semantics> <msub> <mi>z</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> and continuous in variable <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> For this class of functions there is introduced a concept of boundedness of <i>L</i>-index in the direction <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">b</mi> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="bold">L</mi> <mo>:</mo> <msup> <mi mathvariant="double-struck">C</mi> <mi>n</mi> </msup> <mo>&#8594;</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </mrow> </semantics> </math> </inline-formula> is a positive continuous function. We present necessary and sufficient conditions of boundedness of <i>L</i>-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 <i>L</i>-index in direction in any bounded domain and for any continuous function <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mo>:</mo> <msup> <mi mathvariant="double-struck">C</mi> <mi>n</mi> </msup> <mo>&#8594;</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo>.</mo> </mrow> </semantics> </math> </inline-formula>