Parametric representation of univalent functions with boundary regular fixed points

Given a set $\mathfrak S$ of conformal maps of the unit disk $\mathbb D$ into itself that is closed under composition, we address the question whether $\mathfrak S$ can be represented as the reachable set of a Loewner - Kufarev - type ODE $\mathrm{d}w_t/\mathrm{d}t=G_t\circ w_t$, $w_0=\mathsf{id}_{\mathbb D}$, where the control functions $t\mapsto G_t$ form a convex cone $\mathcal M_{\mathfrak S}$. For the set of all conformal $\varphi:\mathbb D\to\mathbb D$ with $\varphi(0)=0$, $\varphi'(0)>0$, an affirmative answer to this question is the essence of Loewner's well-known Parametric Representation of univalent functions [Math. Ann. 89 (1923), 103-121]. In this paper, we study classes of conformal self-maps defined by their boundary regular fixed points and, in part of the cases, establish their Loewner-type representability.Comment: Version 3: Introduction has been modified; Example 3.1 added + other minor correction

Ірраціональні вірування людини як когнітивні утворення. (Irrational beliefs of personality as cognitive formation)

В статті досліджено психологічні особливості такого конструкту, як «ірраціональні вірування людини», охарактеризовано розуміння життєвих труднощів з точки зору когнітивної парадигми, подано ключові ірраціональні вірування сучасної вітчизняної вибірки, обґрунтовано доцільність вживання терміну «вірування» в контексті різних форм дисфункціональних когніцій людини.(An article researches psychological peculiarities of “human irrational beliefs” construct, describes cognitive approach to understanding of life difficulties, gives core irrational beliefs of contemporary national sample, explains necessity of using a construct of ‘beliefs’ in the context of different forms of human dysfunctional cognitions.

Valiron and Abel equations for holomorphic self-maps of the polydisc

We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map $f: \Delta^N \to \Delta^N$ of the polydisc which does not admit fixed points in $\Delta^N$. We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic $f$ and the Abel equation for a parabolic nonzero-step $f$. This is done by studying the canonical Kobayashi hyperbolic semi-model of $f$ and by obtaining a normal form for the automorphisms of the polydisc. In the case of the Valiron equation we also describe the space of all solutions.Comment: A few references are adde

Matching univalent functions and conformal welding

Given a conformal mapping $f$ of the unit disk $\mathbb D$ onto a simply connected domain $D$ in the complex plane bounded by a closed Jordan curve, we consider the problem of constructing a matching conformal mapping, i.e., the mapping of the exterior of the unit disk $\mathbb D^*$ onto the exterior domain $D^*$ regarding to $D$. The answer is expressed in terms of a linear differential equation with a driving term given as the kernel of an operator dependent on the original mapping $f$. Examples are provided. This study is related to the problem of conformal welding and to representation of the Virasoro algebra in the space of univalent functions.Comment: 17 page

Angular and unrestricted limits of one-parameter semigroups in the unit disk

We study local boundary behaviour of one-parameter semigroups of holomorphic functions in the unit disk. Earlier under some addition condition (the position of the Denjoy - Wolff point) it was shown in [M.D.Contreras, S.Diaz-Madrigal and Ch.Pommerenke, Ann. Acad. Sci. Fenn. Math. 29(2004), No.2, 471-488] that elements of one-parameter semigroups have angular limits everywhere on the unit circle and unrestricted limits at all boundary fixed points. We prove stronger versions of these statements with no assumption on the position of the Denjoy - Wolff point. In contrast to many other problems, in the question of existence for unrestricted limits it appears to be more complicated to deal with the boundary Denjoy - Wolff point (the case not covered in [M.D.Contreras, S.Diaz-Madrigal and Ch.Pommerenke, Ann. Acad. Sci. Fenn. Math. 29(2004), No.2, 471-488]) than with all the other boundary fixed points of the semigroup.Comment: Some changes are made in this version: misprints and minor errors are corrected, the information on the grant support is adde