In the paper we investigate slice holomorphic functions F : Cn→C having bounded L-index in a direction, i.e. these functions are entire on every slice {z0 +tb : t∈C} for an arbitrary z0∈Cn and for the fixed direction b∈Cn \ {0}, and (∃m0∈Z+) (∀m∈Z+) (∀z∈Cn) the following inequality holds [wzór], where L : Cn→R+ is a positive continuous function, [wzór] for p≥2. Also, we consider index boundedness in the direction of slice holomorphic solutions of some partial differentia equations with partial derivatives in the same direction. There are established sufficient conditions providing the boundedness of L-index in the same direction for every slie holomorphic solutions of these equations.
In this paper, there are improved sufficient conditions of boundedness of the L-index in a direction for entire solutions of some linear partial differential equations. They are new even for the one-dimensional case and L≡1. Also, we found a positive continuous function l such that entire solutions of the homogeneous linear differential equation with arbitrary fast growth have a bounded l -index and estimated its growth.
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