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2 2019

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Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction

Bandura Andriy, Skaskiv Oleh, Smolovyk Liana

Demonstratio Mathematica

2019

Vol. 52, nr 1

482--489

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.

Analytic functions in the unit ball of bounded L-index in joint variables and of bounded L-index in direction: a connection between these classes

Bandura Andriy, Skaskiv Oleh

Demonstratio Mathematica

2019

Vol. 52, nr 1

82--87

We give negative answer to the question of Bordulyak and Sheremeta for more general classes of entire functions than in the original formulation: Does index boundedness in joint variables for an entire function F imply index boundedness in the variable zj for the function F? This question is addressed for entire functions of bounded L-index in joint variables and entire functions of bounded L-index in direction. We also present a class of analytic functions in the unit ball which has bounded L-index in joint variablesand has unbounded l-index in the variables z1 and z2 for any positive continuous function l : B2→C.