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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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A sharpened form of the inverse function theorem

100%
Elin M., Shoikhet D.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2019
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tom 73
|
nr 2
EN
In this note we establish an advanced version of the inverse function theorem and study some local geometrical properties like starlikeness and hyperbolic convexity of the inverse function under natural restrictions on the numerical range of the underlying mapping.
2
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Coefficient body for nonlinear resolvents

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Elin M., Jacobzon F.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2019
|
tom 73
|
nr 2
EN
This paper is devoted to the study of families of so-called nonlinear resolvents. Namely, we construct polynomial transformations which map the closed unit polydisks onto the coefficient bodies for the resolvent families. As immediate applications of our results we present a covering theorem and a sharp estimate for the Schwarzian derivative at zero on the class of resolvents.
3
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Complex dynamical systems and the geometry of domains in Banach spaces

88%
Elin M., Reich S., Shoikhet D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
We consider semigroups of holomorphic self-mappings on domains in Hilbert and Banach spaces, and then develop a new dynamical approach to the study of geometric properties of biholomorphic mappings. We establish, for example, several flow invariance conditions and find parametric representations of semicomplete vector fields. In order to examine the asymptotic behavior of these semigroups, we use diverse tools such as hyperbolic metric theory and estimates of solutions of generalized differential equations. In addition, we introduce a new method involving admissible upper and lower bounds. Finally, we apply our dynamical approach to obtain several growth and covering theorems for star-like mappings on the open unit balls of Banach and Hilbert spaces.
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On the Rogosinski radius for holomorphic mappings and some of its applications

81%
Aizenberg L., Elin M., Shoikhet D.
Studia Mathematica
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2005
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tom 168
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nr 2
147-158
EN
The well known theorem of Rogosinski asserts that if the modulus of the sum of a power series is less than 1 in the open unit disk: $|∑_{n=0}^{∞} aₙzⁿ| < 1$, |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: $|∑_{n=0}^{k} aₙzⁿ| < 1$, |z| < 1/2, and this radius is sharp. We present a generalization of this theorem to holomorphic mappings of the open unit ball into an arbitrary convex domain. Other multidimensional analogs of Rogosinski's theorem as well as some applications to dynamical systems are considered.
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