AMBIENTUM BIOETHICA BIOLOGIA CHEMIA DIGITALIA DRAMATICA EDUCATIO ARTIS GYMNAST. ENGINEERING EPHEMERIDES EUROPAEA GEOGRAPHIA GEOLOGIA HISTORIA HISTORIA ARTIUM INFORMATICA IURISPRUDENTIA MATHEMATICA MUSICA NEGOTIA OECONOMICA PHILOLOGIA PHILOSOPHIA PHYSICA POLITICA PSYCHOLOGIA-PAEDAGOGIA SOCIOLOGIA THEOLOGIA CATHOLICA THEOLOGIA CATHOLICA LATIN THEOLOGIA GR.-CATH. VARAD THEOLOGIA ORTHODOXA THEOLOGIA REF. TRANSYLVAN
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Autori: ANDRIY BANDURA.
Received 24 December 2021; Accepted 18 January 2022.
pp. 335-350
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In this paper, we investigate analytic solutions of higher order linear non-homogeneous directional differential equations whose coefficients are analytic functions in the unit ball. We use methods of theory of analytic functions in the unit ball having bounded L๐ฟ-index in direction, where L:BnโR+๐ฟ:๐ต๐โ๐ + is a continuous function such that L(z)>ฮฒ|b|1โ|z|๐ฟ(๐ง)>๐ฝ|๐|1โ|๐ง| for all zโBn,๐งโ๐ต๐, bโCnโ{0}๐โ๐ถ๐โ{0} be a fixed direction, ฮฒ>1๐ฝ>1 is some constant. Our proofs are based on application of inequalities from analog of Haymanโs theorem for analytic functions in the unit ball. There are presented growth estimates of their solutions which contains parameters depending on the coefficients of the equations. Also we obtained sufficient conditions that every analytic solution of the equation has bounded L๐ฟ-index in the direction. The deduced results are also new in one-dimensional case, i.e. for functions analytic in the unit disc.
Mathematics Subject Classification (2010): 32W50, 32A10, 32A17.
Keywords: analytic function, analytic solution, slice function, unit ball, directional differential equation, growth estimate, bounded $L$-index in direction
