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We study sufficient conditions of boundedness of  L -index in a direction  b  ∈ ℂ  n   ∖ {0} for analytic solutions in the unit ball of a linear higher order non-homogeneous differential equation with directional derivatives. These conditions are restrictions by the analytic coefficients in the unit ball of the equation. Also we investigate asymptotic behavior of analytic functions of bounded  L -index in the direction and estimate its growth. The results are generalizations of known propositions for entire functions of several variables.

Linear directional differential equations in the unit ball: solutions of bounded L -index