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We prove that if E ⊆ Cn is a Φ-like domain and D ⊆ E is a Φ|D--like domain, then (D, E) is a Runge pair. Certain applications, examples and questions are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we survey recent results obtained by the authors on the preservations of the first elements of (g-) Loewner chains and the Bloch mappings by the Roper-Suffridge type extension operators, the Muir type extension operators and the Pfaltzgraff-Suffridge type extension operators into the mappings on the domains in the complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper we survey recent results on Loewner theory and approximation properties of univalent mappings on the Euclidean unit ball Bn in Cn. We present recent applications of the Andersén-Lempert theorem, concerning the locally uniform approximation of biholomorphic mappings of starlike domains in Cn onto Runge domains by automorphisms of Cn (n ≥ 2), in the study of the following geometric properties of univalent mappings on the Euclidean unit ball Bn: convexity, starlikeness, spirallikeness, parametric representation and embedding into univalent subordination chains. We consider density results and characterizations of these geometric properties through automorphisms of Cn, when n ≥ 2. We also consider density results for families of normalized star-like, spirallike, convex, and mappings with parametric representation on Bn, by smooth quasiconformal diffeomorphisms of Cn. On the other hand, we consider the following question: under which conditions is a normalized univalent mapping on Bn (n ≥ 2), up to a normalized automorphism of Cn, spirallike (or starlike, respectively convex)? This question was first studied by Arosio, Bracci and Wold for convex mappings, by using a smooth boundary assumption, while the authors of the present paper extended their work under lower boundary assumptions, by considering also the case of spirallike mappings. [ABSTRACT FROM AUTHOR]

Abstract Let be the unit ball in with respect to the Euclidean norm. In this paper, we obtain a sufficient condition for a normalized quasiregular mapping to be extended to a quasiconformal homeomorphism of onto itself. In the last section we consider the asymptotical case of this result and we obtain certain applications.

Let Ω ⊂ Cn be a bounded balanced pseudoconvex domain with C¹ plurisubharmonic defining functions. In this paper, we study properties of the Carathéodory family M(Ω), and prove that it is compact with respect to the topology of locally uniform convergence on H(Ω). Then we survey some results in the theory of Loewner chains on Ω, which are extensions to bounded balanced pseudoconvex domains of corresponding results on the Euclidean unit ball Bn in Cn . A special interest in our discussion is played by the family S0 (Ω) of normalized univalent mappings which have parametric representation on Ω. Certain questions and conjectures related to the family S0 (Ω) are also considered. [ABSTRACT FROM AUTHOR]

Let Y be a complex Banach space and let B Y be the open unit ball of Y. In this paper we consider a generalization of the Pfaltzgraff-Suffridge extension operator on bounded symmetric domains in C n , and prove that if B X is a bounded symmetric domain in X = C n , and F n , α is an extension operator which maps normalized locally biholomorphic mappings on B X to locally biholomorphic mappings on D α , where D α ⊆ B X × B Y is a certain domain with B X × { 0 } ⊂ D α , then F n , α extends the first elements of Loewner chains from B X to the first elements of Loewner chains on D α , when α ≥ n / (2 c (B X)) , where c (B X) is a constant defined by the Bergman metric on B X . Particular cases are also obtained. Next, we prove that normalized locally univalent I-Bloch mappings, which have finite trace order on B X , are mapped into R-Bloch mappings on Ω α by the operator F n , α when α ≥ 1 / 2 , where Ω α ⊂ X × Y is a bounded balanced convex domain such that B X × { 0 } ⊂ Ω α ⊆ D α . In particular, the normalized bounded convex mappings on B X are mapped into R-Bloch mappings on Ω α under the extension operator F n , α . [ABSTRACT FROM AUTHOR]

Let BX be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*-triple X=(Cn,‖·‖X). In this paper, we give a definition of α-Bloch mappings on BX which is a generalization of α-Bloch functions on the unit disc in C. This definition is new in the case of the Euclidean unit ball Bn in Cn. We generalize Bonk’s distortion theorem to α-Bloch mappings on BX. As an application, we give a lower bound of the Bloch constant for α-Bloch mappings on BX. [ABSTRACT FROM AUTHOR]

Let ...(Bn) be the family of normalized univalent mappings on the Euclidean unit ball Bn in Cn, which have generalized parametric representation with respect to time-dependent operators A ∈ Ã, where à is a family of measurable mappings from [0, ∞) into L(Cn) with some particular properties. Also, let ... be the time-T-reachable family of the control system C([T0, T], (...). where A ∈ A and T0 ≥ 0. In this paper we obtain certain convergence results for the families ...(Bn) and ... with respect to the Hausdorff metric p on H(Bn). These results may be seen as dominated convergence type theorems for time-dependent operators A ∈ Ã. In particular, we obtain related convergence results for the family ...(Bn) (resp. for the family ŜA(Bn) of mappings with A-parametric representation on Bn (resp. of spirallike mappings on Bn with respect to A), in the case that A ∈ L(Cn) is a linear operator with k+ (A) < 2m(A), where k+(A) is the Lyapunov index of A and m(A) = min||x||=1 R{A(z), z). We also obtain a convergence result for the Carathéodory family NA, where m(A) > 0. Finally, we obtain some sufficient conditions related to A ∈ Ã, which yield the equality ...(Bn) = S0(Bn), for all t ≥ 0, where S0(Bn) is the family of normalized univalcnt mappings with usual parametric representation on Bn. Certain consequences arc also provided. [ABSTRACT FROM AUTHOR]

Let $$A\in L({\mathbb {C}}^n)$$ be a linear operator such that $$k_+(A)<2m(A)$$ , where $$k_+(A)$$ is the upper exponential index of $$A$$ and $$m(A)=\min \{\mathfrak {R}\langle A(z),z\rangle :\Vert z\Vert =1\}$$ . In this paper we are concerned with variations of $$A$$ -normalized univalent subordination chains on the Euclidean unit ball $$\mathbb {B}^n$$ in $$\mathbb {C}^n$$ . We also obtain growth results for the generating vector fields and the transition mappings associated with $$A$$ -normalized univalent subordination chains $$f(z,t)$$ that satisfy some regularity assumptions. Further, we give examples of normalized biholomorphic mappings on $$\mathbb {B}^n$$ which are not extreme/support points for the compact family $$S_A^0(\mathbb {B}^n)$$ of mappings with $$A$$ -parametric representation on $$\mathbb {B}^n$$ . Next, we consider the (normalized) time- $$\log M$$ -reachable family $$\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^n},{\fancyscript{N}}_A)$$ generated by the Carathéodory mappings, where $$M\in (1,\infty )$$ and $$k_+(A)<2m(A)$$ . We prove a result related to the support points $$f$$ of the family $$\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^n},{\fancyscript{N}}_A)$$ and the associated $$A$$ -univalent subordination chains $$f(z,t)$$ such that $$f=f(\cdot ,0)$$ . In particular, if $$f\in \tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^n},{\fancyscript{N}}_A)$$ , then $$f\not \in (\mathrm{supp}\, S_A^0(\mathbb {B}^n)\cup \mathrm{ex}\, S_A^0(\mathbb {B}^n))$$ . In the last part of the paper, using some ideas based on a shearing process that has been introduced recently by F. Bracci, we obtain a sharp estimate for $$\Big |\frac{\partial ^2 f_1}{\partial z_2^2}(0)\Big |$$ , when $$f=(f_1,f_2)\in S_A^0(\mathbb {B}^2)$$ and $$A$$ is a diagonal matrix whose diagonal elements are $$\lambda $$ and $$1, \lambda \in [1,2)$$ . This result allows us to construct an example of a bounded support mapping for the family $$S_A^0(\mathbb {B}^2)$$ , but which does not belong to any reachable family $$\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^2},{\fancyscript{N}}_A)$$ , for all $$M>1$$ . This example provides a basic difference between the theory of bounded univalent mappings on the unit disc $$U$$ and that on the unit ball $$\mathbb {B}^n$$ in $$\mathbb {C}^n, n\ge 2$$ . Finally, we construct an example of a support point $$F_A^M$$ for the reachable family $$\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^2},{\fancyscript{N}}_A)$$ , but which is not a support point/extreme point for the family $$S_A^0(\mathbb {B}^2)$$ , and we conjecture that the mapping $$F_A^M$$ is also an extreme point for the family $$\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^2},{\fancyscript{N}}_A)$$ . [ABSTRACT FROM AUTHOR]

We introduce a family of natural normalized Loewner chains in the unit ball, which we call 'geräumig'-spacious-which allow us to construct, by means of suitable variations, other normalized Loewner chains which coincide with the given ones from a certain time on. We apply our construction to the study of support points, extreme points, and time- $$\log M$$ -reachable mappings in the class $$S^0$$ of mappings admitting parametric representation. [ABSTRACT FROM AUTHOR]