Non-homogeneous directional equations: Slice solutions belonging to functions of bounded $L$-index in the unit ball

Abstract

For a given direction $ b\in\mathbb{C}^n\setminus\{0\}$ we study non-homogeneous directional linear higher-order equations whose all coefficients belong to a class of joint continuous functions which are holomorphic on intersection of all directional slices with a unit ball. Conditions are established providing boundedness of $L$-index in the direction with a positive continuous function $L$ satisfying some behavior conditions in the unit ball. The provided conditions concern every solution belonging to the same class of functions as the coefficients of the equation. Our considerations use some estimates involving a directional logarithmic derivative and distribution of zeros on all directional slices in the unit ball.\looseness+1

Keywords

• bounded index
• bounded $l$-index in direction
• slice function
• holomorphic function
• directional differential equation
• bounded $l$-index
• directional derivative
• unit ball