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

Abstract

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.

Keywords

• bounded l-index in direction
• entire function of several variables
• bounded l-index in joint variables
• analytic function in the unit ball
• existence theorem