Analytic functions in the unit ball of bounded L-index in joint variables and of bounded 𝐿-index in direction: a connection between these classes

• Main Authors: Bandura Andriy, Skaskiv Oleh
• Format: Article
• Language: English
• Published: De Gruyter 2019-02-01
• Series: Demonstratio Mathematica
• Subjects: bounded l-index in direction; entire function of several variables; bounded l-index in joint variables; analytic function in the unit ball; existence theorem
• Online Access: https://doi.org/10.1515/dema-2019-0008
• ISSN: 2391-4661

We give negative answer to the question of Bordulyak and Sheremeta for more general classes of entire functions than in the original formulation: Does index boundedness in joint variables for an entire function F imply index boundedness in the variable zj for the function F? This question is addressed for entire functions of bounded L-index in joint variables and entire functions of bounded L-index in direction. We also present a class of analytic functions in the unit ball which has bounded L-index in joint variables and has unbounded l-index in the variables z1 and z2 for any positive continuous function l : B2 → C.