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1

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

Bandura Andriy, Skaskiv Oleh, Smolovyk Liana

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

482--489

EN

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.

2

Defective functions of meromorphic functions in the unit disc

Ciechanowicz E.

Journal of Applied Analysis

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2016

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Vol. 22, nr 1

15--26

EN

Let f be a meromorphic function in the unit disc and (aν)kν=1 a set of distinct meromorphic functions small with respect to f. An analogue of the second main theorem for f and (aν)kν=1 is given. Upper limits for the sum of defects of an admissible meromorphic function and an admissible holomorphic function follow. For meromorphic and holomorphic functions in the unit disc and their small functions the analogues of Ullrich's theorem are presented.

3

On certain functions with positive real part

Sendrutiu R.

Journal of Mathematics and Applications

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2010

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Vol. 32

85-90

4

Transformation of the Vistula Lagoon onto a canonical domain

Prosnak W. J., Cześnik P. P.

Oceanologia

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2001

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No. 43 (2)

169-200

EN

The paper deals with the conformal mapping of finite, plane, simply connected domains, representing oceans, lakes, estuaries, bays, lagoons, and other natural water bodies of this kind. As a rule, they are bounded by geometrically complex shorelines. The partial differential problems investigated in Oceanology and posed in such domains have turned out to be difficult to solve for at least three reasons. They follow on from the mathematical properties of the differential equations governing such problems, from the just-mentioned geometrical complexity of the domains of solution, and from the sensitivity of the solutions to boundary conditions. In view of the last reason the contours admitted as boundaries of the domains of the solution ought to be as close to the real shorelines as possible. The obviously inaccurate approximation of the shorelines by "staircases", which appears rather often (cf. Catewicz & Jankowski 1983, Lin & Chandler-Wilde 1996) as a consequence of applying finite difference methods to the solution of the partial differential problems, raises serious doubts from the point of view of Numerical Fluid Mechanics. It is recalled in the paper that such inaccuracies are not unavoidable: that complicated plane domains can be transformed accurately by means of properly applied conformal mapping onto regular, canonical domains - in particular, onto discs or squares. Such a transformation is demonstrated on the rather difficult example of the Vistula Lagoon. The transformation begins with the decomposition of the domain into five plane subdomains, each one of which is eventually transformed onto a disc. Every such result is arrived at quite independently of the remaining subdomains, by means of a set of properly selected consecutive mappings. Hence, the final canonical domain consists in this case of a system of five discs which, however, within the framework of this differential problem, have to be treated as interconnected. The interconnections involve images of four segments of straight lines, separating the original subdomains. The transformations and the resulting canonical domain presented in the paper are intended to be applied to the solution of certain hydrodynamical problems concerning the Vistula Lagoon, which will be published elsewhere.

5

On factorization of polynomials with holomorphic coefficients

Frontczak M., Skibiński A., Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

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1998

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Vol. 46, no 1

39-54

EN

In the paper some equivalent conditions to pointwise reducibility of polynomials with holomorphic coefficients (in an arbitrary connected set) are given.