Non-homogeneous directional equations: Slice solutions belonging to functions
of bounded $L$-index in the unit ball
Andriy Bandura, Tetyana Salo, Oleh Skaskiv
Received September 8, 2022. Published online April 12, 2023.
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.
Keywords: bounded index; bounded $L$-index in direction; slice function; holomorphic function; directional differential equation; bounded $l$-index; directional derivative; unit ball
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Affiliations: Andriy Bandura (corresponding author), Ivano-Frankivsk National Technical University of Oil and Gas, Karpatska 15, 76019 Ivano-Frankivsk, Ukraine, e-mail: andriykopanytsia@gmail.com; Tetyana Salo, Lviv Polytechnic National University, Stepana Bandery St. 12, 79000 Lviv, Ukraine, e-mail: tetyan.salo@gmail.com; Oleh Skaskiv, Ivan Franko National University of Lviv, Universytetska St. 1, 79000 Lviv, Ukraine, e-mail: olskask@gmail.com
