arXiv subjects

Fiana Jacobzon

Publications and source records attributed to Fiana Jacobzon.

16 recordsLinked to original sources

arXiv2025

On one filtration of holomorphic functions

In this work we consider a family of function classes constructed by means of the Gauss hypergeometric function $_2F_1(1,1;2;z) =-\frac{\log(1-z)}{z}$. We demonstrate that this family, in fact, constitutes classes of analytic functions subject to prescribed constraints on their derivatives. For these classes we obtain some geometric characteristics, including sharp coefficient estimates. Moreover, we show that this family naturally provides a filtration of infinitesimal generators, and investigate the corresponding dynamical behavior of the associated semigroups. It is interesting that this filtration links to the Ma--Minda starlike functions.

math.CV↗

arXiv2024

On the hypergeometric function and families of holomorphic functions

In this work, we examine one two-parameter family of sets consisting of functions holomorphic in the unit disk, previously investigated by several mathematicians. We focus on the set-theoretic properties of this family, identify the general form of filtrations within it, and discover that it is not a lattice. This insight motivates us to introduce a refined concept of quasi-infima and quasi-suprema, and to establish their complete description. Unexpectedly, some new properties of the Gauß hypergeometric function play a crucial role in our investigation.

math.CV↗

arXiv2024

Multidimensional analogs of the Fekete--Szegö functional

In this paper we introduce the Fekete--Szegö type mapping in the open unit ball of a complex Banach space. % and study its geometric and analytical properties. All previously studied modifications of the Fekete--Szegö functional are either special cases or `components' of the mapping we introduce. The study involves the examination of transforms of the Fekete--Szegö mapping under specific transformations applied to given holomorphic mappings. We show that for a mapping $f$, the third order Fréshet derivative of the inverse mapping $f^{-1}$ and of elements of the semigroup generated by $f$ can be expressed in terms of the Fekete--Szegö mapping. Estimates of the Fekete--Szegö mapping over some subclasses of semigroup generators and of starlike mappings are also presented.

math.CV↗

arXiv2024

Nonlinear resolvents in the unit disk: geometry and dynamics

In this paper we present a unified approach to the study of geometric and dynamic properties of nonlinear resolvents of holomorphic generators. The idea is to apply the distortion theorem we have established. This method allows us to find order of spirallikeness and of strong starlikeness of resolvents and remove all the restrictions for resolvents to admit quasiconformal extension to the complex plane $\C$. In addition, we use this method to establish the uniform convergence of the resolvent family on the whole unit disk and obtain some characteristics of semigroups generated by these resolvents.

math.CV↗

arXiv2023

Survey on filtrations (parametric embeddings) of infinitesimal generators

This work is devoted to the so-called filtration theory of semigroup generators in the unit disk. It should be noted that numerous filtrations studied to nowdays have been introduced for different purposes and considered from different points of view. So, our aim is to summarize the known facts, to present common and distinct properties of filtrations as well as to study some new filtrations with an emphasis on their connection with geometric function theory and the dynamic features of semigroups generated by elements of different filtration families. Among the dynamic properties, we mention the uniform convergence on the unit disk and the sectorial analytical extension of semigroups with respect to their parameter. We also solve the Fekete--Szegö problem over various filtration classes, as well as over non-linear resolvents.

math.CV↗

arXiv2022

Geometric features of nonlinear resolvents in the unit disk

We study nonlinear resolvents of holomorphic generators of one-parameter semigroups acting in the open unit disk. The class of nonlinear resolvents can be studied in the framework of geometric function theory because it consists of univalent functions. In this paper we establish distortion and covering results, find order of starlikeness and of strong starlikeness of resolvents. This provides that any resolvent admits quasiconformal extension to the complex plane $\C$. In addition, we obtain some characteristics of semigroups generated by these resolvents.

math.CV↗

arXiv2022

The Fekete--Szegö problem for spirallike mappings and non-linear resolvents in Banach spaces

We study the Fekete--Szegö problem on the open unit ball of a complex Banach space. Namely, the Fekete--Szegö inequalities are proved for the class of spirallike mappings relative to an arbitrary strongly accretive operator, and some of its subclasses. Next, we consider families of non-linear resolvents for holomorphically accretive mappings vanishing at the origin. We solve the Fekete--Szegö problem over these families.

math.CV↗

arXiv2022

Note on the Fekete--Szegö problem for spirallike mappings in Banach spaces

In this note we present a remark on the paper "On the coefficient inequalities for a class of holomorphic mappings associated with spirallike mappings in several complex variables" by Y.~Lai and Q.~Xu \cite{LX} published recently in the journal {\it Results in Mathematics}. We show that one of the theorems in \cite{LX} concerning the finite-dimensional space $\mathbb{C}^n$ is a direct consequence of another one, so it does not need an independent proof. Moreover, we prove that a sharp norm estimate on the Fekete--Szegö functional over spirallike mappings in a general Banach space can be deduced from a result in \cite{LX}.

math.CV↗

arXiv2021

The Fekete--Szegö problem and filtration of generators

In this paper we study an interpolation problem involving the Fekete--Szegö functional. It turns out that this problem links to the so-called filtration of infinitesimal generators. We introduce new filtration classes using the non-linear differential operator \[α\frac{f(z)}{z}+β\frac{zf'(z)}{f(z)}+(1-α-β)\left(1+\frac{zf''(z)}{f'(z)}\right)\] and establish certain properties of these classes. Sharp upper bounds of the modulus of the Fekete--Szegö functional over some filtration classes are found. We also present open problems for further study.

math.CV↗

arXiv2021

Estimates on some functionals over non-linear resolvents

Estimation of linear and quadratic functionals over different classes of univalent functions is one of the classical problems in geometric function theory. In this paper we solve the problem over some classes of so-called non-linear resolvents, which arise as a fruitful tool in dynamic systems. Sharp estimates on early Taylor coefficients and the Fekete--Szegö functional are established.

math.CV↗

arXiv2020

Families of inverse functions: coefficient bodies and the Fekete--Szegö problem

In this paper we establish the coefficient bodies for a wide class of families of inverse functions. We also completely describe those functions that provide boundary points of that bodies in small dimensions. As an application we get sharp bounds for Fekete--Szegö functionals over some classes of functions defined by quasi-subordination as well as over classes of their inverses. As a biproduct we derive a formula for ordinary Bell polynomials that seems to be new.

math.CV↗

arXiv2019

Linearization of holomorphic semicocycles in Banach spaces

We consider holomorphic semicocycles on the open unit ball in a Banach space taking values in a Banach algebra. We establish criteria for a semicocycle to be linearizable, that is, cohomologically equivalent to one independent of the spatial variable.

math.DS↗

arXiv2018

Continuous and holomorphic semicocycles in Banach spaces

We study some fundamental properties of semicocycles over semigroups of self-mappings of a domain in a Banach space. We prove that any semicocycle over a jointly continuous semigroup is itself jointly continuous. For semicocycles over semigroups which have generator, we establish a sufficient condition for differentiablity with respect to the time variable, and hence for the semicocycle to satisfy a linear evolution problem, giving rise to the notion of `generator' of a semicocycle. Bounds on the growth of a semicocycle with respect to the time variable are given in terms of this generator. Special consideration is given to the case of holomorphic semicocycles, for which we prove an exact correspondence between certain uniform continuity properties of a semicocyle and boundedness properties of its generator.

math.FA↗

arXiv2017

Non-commutative holomorphic semicocycles

This paper studies holomorphic semicocycles over semigroups in the unit disk, which take values in an arbitrary unital Banach algebra. We prove that every such semicocycle is a solution to a corresponding evolution problem. We then investigate the linearization problem: which semicocycles are cohomologous to constant semicocycles? In contrast with the case of commutative semicocycles, in the non-commutative case non-linearizable semicocycles are shown to exist. Simple conditions for linearizability are derived and are shown to be sharp.

math.CV↗

arXiv2015

Analyticity of semigroups on the right half-plane

This paper is devoted to the study of semigroups of composition operators and semigroups of holomorphic mappings. We establish conditions under which these semigroups can be extended in their parameter to sector given a priori. We show that the size of this sector can be controlled by the image properties of the infinitesimal generator, or, equivalently, by the geometry of the so-called associated planar domain. We also give a complete characterization of all composition operators acting on the Hardy space $H^p$ on the right half-plane.

math.CV↗

arXiv2013

Parabolic type semigroups: asymptotics and order of contact

We study the asymptotic behavior of parabolic type semigroups acting on the unit disk as well as those acting on the right half-plane. We use the asymptotic behavior to investigate the local geometry of the semigroup trajectories near the boundary Denjoy--Wolff point. The geometric content includes, in particular, the asymptotes to trajectories, the so-called limit curvature, the order of contact, and so on. We then establish asymptotic rigidity properties for a broad class of semigroups of parabolic type.

math.CV↗