Hidetaka Hamada

Kyushu Sangyo University · Faculty of Engineering

The purpose of this paper is to develop some methods to study (Fejér-)Riesz type inequalities, Hardy–Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some sharp Riesz type inequalities of pluriharmonic functions on bounded symmetric domains. The obtained r...

The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Banach spaces. Initially, we extend the classical Schwarz lemmas of holomorphic mappings to Banach spaces, and then we apply these extensions to establish a sharp Bloch type theorem for pluriharmonic...

The main purpose of this paper is to discuss Hardy type spaces and Bergman type classes of complex-valued harmonic functions. We first establish a Hardy-Littlewood type theorem on complex-valued harmonic functions. Next, the relationships between the Bergman type classes and the Hardy type spaces of complex-valued harmonic functions or the relation...

The main aim of this paper is to investigate the Hardy-Littlewood type Theorem and the Heinz type inequality on functions induced by a differential operator. We first prove a more general Hardy-Littlewood type theorem for the Dirichlet solution of a differential operator which depends on $\alpha >0$ over the unit ball $\mathbb{B}^n$ of $\mathbb{R}^...

Let BH$\mathbb {B}_H$ be the unit ball of a complex Hilbert space H. First, we give a Bohr's inequality for the holomorphic mappings with lacunary series with values in complex Hilbert balls. Next, we give several results on Bohr's inequality for pluriharmonic mappings with values in ℓ2. Note that the Bohr phenomenons that we have obtained are comp...

The main purpose of this paper is to develop some methods to study the composition operators between harmonic Lipschitz type spaces. Some characterizations of boundedness and w-compactness of composition operators between the harmonic Lipschitz type spaces will be given. Consequently, the obtained results improve and extend some corresponding known...

The main purpose of this paper is to develop some methods to investigate equivalent norms and Hardy-Littlewood type Theorems on Lipschitz type spaces of analytic functions and complex-valued harmonic functions. Initially, some characterizations of equivalent norms on Lipschitz type spaces of analytic functions and complex-valued harmonic functions...

In the first part of this paper, we will give the Fekete–Szegö inequality for various subfamilies of spirallike mappings of type β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d...

In the first part of this paper, we give generalizations of the Fekete–Szegö inequalities for quasiconvex mappings F of type B and the first elements F of g-Loewner chains on the unit ball of a complex Banach space, recently obtained by H. Hamada, G. Kohr and M. Kohr. We obtain the Fekete–Szegö inequalities using the norm under the restrictions on...

The purpose of this paper is to develop some methods to study Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some sharp Riesz type inequalities of pluriharmonic functions on bounded symmetric domains. The obtained results e...

The main purpose of this paper is to investigate a Fej\'er-Riesz type inequality and composition operators of high dimensional cases. Initially, we establish a Fej\'er-Riesz type inequality on pluriharmonic functions. Furthermore, by using weights, we develop some methods to study the composition operators from harmonic Bloch type spaces to pluriha...

The main purpose of this paper is to discuss Hardy type spaces, Bloch type spaces and the composition operators of complex-valued harmonic functions. We first establish a sharp estimate of the Lipschitz continuity of complex-valued harmonic functions in Bloch type spaces with respect to the pseudo-hyperbolic metric, which gives an answer to an open...

The main purpose of this paper is to discuss Hardy type spaces, Bloch type spaces and the composition operators of complex-valued harmonic functions. We first establish a sharp estimate of the Lipschitz continuity of complex-valued harmonic functions in Bloch type spaces with respect to the pseudo-hyperbolic metric, which gives an answer to an open...

In this paper, we will give the Fekete-Szegö inequality for the mappings f in various subclasses of normalized univalent mappings which are the first elements of g-Loewner chains on the unit disc U in C and also on the unit ball B of a complex Banach space. As an application, we give the estimation of the third coefficient for f under the condition...

We prove that if $E\subseteq \Cn$ is a $\Phi$-like domain and $D\subseteq E$ is a $\Phi\big|_D$-like domain, then $(D,E)$ is a Runge pair. Certain applications, examples and questions are also provided.

"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 spa...

The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in Rn into the unit ball of the real Minkowski space. Next, we give several sharp Sc...

Let \({\mathbb {B}}_X\) be a bounded symmetric domain realized as the open unit ball \({\mathbb {B}}_X\) of a finite dimensional JB*-triple X. In this paper, we continue the work related to the composition operator \(C_{\varphi }\) between Bloch-type spaces, where \(\varphi \) is a holomorphic mapping from \({\mathbb {B}}_X\) into the unit polydisc...

Let BX be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*‐triple X. In this paper, we obtain a rigidity theorem at the boundary for holomorphic mappings from a balanced domain G in a complex Banach space E into BX. We also obtain a rigidity theorem at the boundary for holomorphic self‐mappings of BX. Our result...

The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Banach spaces. Initially, we extend the classical Schwarz lemmas of holomorphic mappings to Banach spaces, and then we apply these extensions to establish a sharp Bloch type theorem for pluriharmonic...

In this paper, we first give a coefficient inequality for holomorphic functions on the unit disc \({\mathbb {U}}\) in \({\mathbb {C}}\) which are subordinate to a holomorphic function p on \({\mathbb {U}}\) with \(p'(0)\ne 0\). Next, as applications of this theorem, we will give the Fekete-Szegö inequality for subclasses of normalized starlike mapp...

In this paper, we will obtain a new result on the Bohr phenomenon for analytic functions f which are subordinate to starlike functions g∈S⁎(ϕ), where ϕ satisfies Ma-Minda conditions and the coefficients of ϕ are non-negative. Next, we will obtain a new result on the Bohr phenomenon for analytic functions f which are subordinate to convex functions...

Let \({\mathbb {B}}_X\) and \({\mathbb {B}}_Y\) be bounded symmetric domains realized as the unit balls of \(\hbox {JB}^*\)-triples X and Y, respectively. In this paper, we generalize the Landau theorem to holomorphic mappings on \({\mathbb {B}}_X\) using the Schwarz–Pick lemma for holomorphic mappings on \({\mathbb {B}}_X\). Next, we give a necess...

In this paper we study various properties of nonlinear resolvents of holomorphic mappings in the Carathéodory family M(Bn), where Bn is the Euclidean unit ball in Cn. First, we prove certain characterizations of inverse Loewner chains f(z,t)=e−∫0ta(τ)dτz+⋯ on Bn×[0,∞), where a:[0,∞)→C is a locally Lebesgue integrable function such that ℜa(t)>0 for...

The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions with values in the unit ball of the Minkowski space. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in $\mathbb{R}^n$ into the unit...

In this paper, we obtain a boundary Schwarz lemma for C1 (pluriharmonic, holomorphic) mappings from the unit polydisc Un in Cn to irreducible bounded symmetric domains realized as the unit ball BX of an N‐dimensional simple JB*‐triple X. In particular, we obtain a version of the boundary Schwarz lemma for C1 (pluriharmonic, holomorphic) mappings fr...

In this paper, we prove a Schwarz lemma at the boundary for holomorphic mappings f between Hilbert balls, and obtain related consequences. Especially, we obtain estimations of ∥Df(z0)∥ on the holomorphic tangent space for holomorphic mappings f or for homogeneous polynomial mappings f between Hilbert balls. Next, we prove the boundary rigidity theo...

Let Y be a complex Banach space and let r≥1. In this paper, we are concerned with an extension operator Φα,β that provides a way of extending a locally univalent function f on the unit disc U to a locally biholomorphic mapping F∈H(Ωr), where Ωr={(z1,w)∈C×Y:|z1|2+‖w‖Yr<1}. We prove that if f can be embedded as the first element of a g-Loewner chain...

Let \(n\ge 2\) and let \(A\in L({\mathbb {C}}^n)\) be such that \(k_+(A)<2m(A)\). In this paper, we prove that if \(F:{\mathbb {B}}^n\rightarrow {\mathbb {C}}^n\) is a normalized biholomorphic mapping such that \(F({\mathbb {B}}^n)\) is a bounded strictly pseudoconvex domain with \(C^2\) boundary, then \(\overline{F({\mathbb {B}}^n)}\) is polynomia...

In this paper, we study some extremal problems for the family \(S_g^0(\mathbb{B}_X)\) of normalized univalent mappings with g-parametric representation on the unit ball \(\mathbb{B}_X\) of an n-dimensional JB*-triple X with r ⩾ 2, where r is the rank of X and g is a convex (univalent) function on the unit disc \(\mathbb{U}\), which satises some nat...

In this paper we study some extremal problems for the family $S_g^0(\mathbb{B}_X)$ of normalized univalent mappings with $g$-parametric representation on the unit ball $\mathbb{B}_X$ of an $n$-dimensional JB$^*$-triple $X$ with $r\geq 2$, where $r$ is the rank of $X$ and $g$ is a convex (univalent) function on the unit disc $\mathbb{U}$, which sati...

Let Y be a complex Banach space and let BY 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 Cn, and prove that if BX is a bounded symmetric domain in X=Cn, and Fn,α is an extension operator which maps normalized locally biholomorphic mappings on BX...

Let BX be a homogeneous unit ball in X = ℂⁿ. In this paper, we generalize Bonk’s distortion theorem to Bloch mappings on BX. As an application, we give a lower bound of the Bloch constant.

We generalize a number of finite dimensional results on Bloch functions to infinite dimensional bounded symmetric domains. In particular, we characterize the Bloch space as well as the little Bloch space of a Hilbert ball, and give one sufficient and several necessary conditions for a composition operator on a Bloch space to be an isometry. We also...

Let B X be a bounded symmetric domain realized as the open unit ball B X of a finite dimensional JB*-triple X. In this paper, we continue the work related to α-Bloch mappings on B X . We first show that α-Bloch spaces on B X are complex Banach spaces. Next, we give sufficient conditions for the composition operator from the α-Bloch space into the β...

Let n≥2 and let [Formula presented] be such that m(A)>0. In this paper, we use a variational result for A-normalized univalent subordination chains, to deduce that every normalized univalent mapping which has A-parametric representation on [Formula presented] can be approximated locally uniformly on [Formula presented] by mappings which have A-para...

In this paper, we prove a Schwarz lemma at the boundary for holomorphic self-mappings f of finite dimensional irreducible bounded symmetric domains without assuming the boundary regularity of f. Our result generalizes the previous results obtained for holomorphic self-mappings f of the Euclidean unit ball, or of the classical Cartan domains of type...

In this paper we are concerned with extremal problems for mappings with g-parametric representation on the unit polydisc \(\mathbb {U}^2\) of \(\mathbb {C}^2\), where g is a univalent holomorphic function on the unit disc \(\mathbb {U}\) such that g(0) = 1, and which satisfies some natural conditions. In the first part of the paper, we obtain certa...

In this paper, we will give coefficient conditions for mappings of the form f(z) = z/(1+Σ∞k=1 bkz1k) to be starlike or convex on the Euclidean unit ball B in ℂⁿ. Our results give concrete examples of strongly starlike mappings of order a, starlike mappings of order a and convex mappings on B.

Let B be the unit ball in ℂ n with respect to an arbitrary norm on ℂ n . In this paper, we give a necessary and sufficient condition that a Loewner chain f(z,t), such that {e -t f(z,t)} t≥0 is a normal family on B, is k-fold symmetrical. As a corollary, we give a necessary and sufficient condition that a normalized locally biholomorphic mapping on...

In this paper, we will consider classes of subordinations involving partial derivatives of holomorphic mappings in complex Banach spaces.

In this paper, we will consider some sufficient conditions for locally biholomorphic mappings defined in the unit ball in complex Banach spaces to be biholomorphic and to have Φ-like images. As a corollary, we obtain some sufficient condition for locally biholomorphic mappings to be starlike mappings.

J. Agler and N. J. Young [Bull. Lond. Math. Soc. 33, 175–186 (2001; Zbl 1030.32011)] obtained a Schwarz lemma for the symmetrized bidisc. Their proof uses an earlier result of them whose proof is operator-theoretic in nature. They posed the question to give an elementary proof of the Schwarz lemma for the symmetrized bidisc. In this paper, we give...

V. Pescar [Demonstr. Math. 33, No. 1, 51–54 (2000; Zbl 0953.30008); Indian J. Pure Appl. Math. 31, No. 8, 975–978 (2000; Zbl 0962.30005)] investigated the univalence of certain integral operators. We will show that the results are obtained by the Schwarz lemma. We will also give some generalizations.

Let $\mathbb{B}$ be the unit ball of a complex Banach space $X$. In this paper, we will generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball $\mathbb{B}$ by using the radial derivative. Next, we define an extended Ces\`{a}ro operator $T_{\varphi}$ with holomorphic symbol $\varphi$ and characterize those $\varphi$...

Let \({\mathbb {B}}_X\) be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*-triple \(X=({\mathbb {C}}^n, \Vert \cdot \Vert _X)\). In this paper, we give a definition of \(\alpha \)-Bloch mappings on \({\mathbb {B}}_X\) which is a generalization of \(\alpha \)-Bloch functions on the unit disc in \({\mathbb {C}}\)...

In this paper, we give a simple proof for the boundary Schwarz lemma for pluriharmonic mappings between Euclidean unit balls. We also give some generalization to C¹-mappings between domains with smooth boundaries.

In this paper we consider support points for the family of mappings with g-parametric representation on the Euclidean unit ball in , where g is a univalent function on the unit disc in , which satisfies certain natural assumptions. We shall use the shearing process recently introduced by Bracci, to prove the existence of bounded support points for...

In this paper we survey various results concerning extremal problems related to Loewner chains, the Loewner differential equation, and Herglotz vector fields on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\). First, we survey recent results related to extremal problems for the Carathéodory families \({\mathcal M}\) and \({\mathcal N...

In this paper we survey recent results related to extremal problems for the family \(\widetilde {S}^t_A(\mathbb {B}^n)\) of normalized univalent mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\), which have generalized parametric representation with respect to time-dependent operators \(A\in \skew 4\widetilde {\mathcal {A}}...

Let \(\mathbb {B}_X\) be a bounded symmetric domain realized as the open unit ball of a JB*-triple X. In this paper, we characterize the bounded weighted composition operators from the Hardy space \(H^{\infty }(\mathbb {B}_X)\) into the Bloch space on \(\mathbb {B}_X\). We also give estimates on the operator norm. The lower estimate is an improveme...

Let $n\geq 2$. In this paper, we obtain approximation properties of various families of normalized univalent mappings $f$ on the Euclidean unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$ by automorphisms of $\mathbb{C}^n$ whose restrictions to $\mathbb{B}^n$ have the same geometric property of $f$. First, we obtain approximation properties of spirallike...

We introduce and characterize Bloch functions on bounded symmetric domains, which may be infinite dimensional, by extending several well-known equivalent conditions for Bloch functions on the open unit disc U in . We also generalize a number of results concerning Bloch functions on U to bounded symmetric domains. Given a holomorphic mapping φ betwe...

We generalize Bonk's distortion theorem on the unit disc in the complex plane to locally biholomorphic mappings on finite dimensional bounded symmetric domains. As an application, we obtain a lower bound for the Bloch constant for various classes of locally biholomorphic Bloch mappings.

In this paper we are concerned with the family (Formula presented.) ((Formula presented.)) of normalized biholomorphic mappings on the Euclidean unit ball (Formula presented.) in (Formula presented.) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators (Formula presented.), where (Formula presente...

Let S~At(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 ã is a family of measurable mappings from [0,∞) into L(Cⁿ) with some particular properties. Also, let R~T(idBⁿ,(NA(t))t∈[T0, T]) be the time-T-reachab...

In this paper, we obtain a sufficient condition for pluriharmonic mappings on the Euclidean unit ball to be univalent, sense-preserving, quasiconformal and bi-Lipschitz diffeomorphisms on and to have linearly connected images. Also, we give a sufficient condition for pluriharmonic mappings on to have quasiconformal extensions to . Next, we generali...

Let (Formula presented.) be a linear operator such that (Formula presented.), where (Formula presented.) is the upper exponential index of (Formula presented.) and (Formula presented.). In this paper we are concerned with variations of (Formula presented.)-normalized univalent subordination chains on the Euclidean unit ball (Formula presented.) in...

In this paper, we will show that any domain D in C-n which is spirallike with respect to a linear operator A, where m(A) > 0, is Runge. We also show the local uniform approximation of biholomorphic mappings on a spirallike domain D with respect to A, where k(+) (A) < 2m(A), by automorphisms of C-n. Finally, as an application of the above result, we...

Let X be a complex Banach space with the unit ball B . The family MM is a natural generalization to complex Banach spaces of the well-known Carathéodory family of functions with positive real part on the unit disc. We consider subfamilies MgMg of MM depending on a univalent function g . We obtain growth theorems and coefficient bounds for holomorph...

Let X be a reflexive complex Banach space with the unit ball B. In the first part of the paper, we survey various growth and coefficient bounds for mappings in the Carathéodory family \(\mathcal{M}\), which plays a key role in the study of the generalized Loewner differential equation. Then we consider recent results in the theory of Loewner chains...

For a linear operator (Formula presented.) be the upper exponential index of A and let (Formula presented.). Under the assumption (Formula presented.), we consider the family SA0(Bn) of mappings which have A-parametric representation on the Euclidean unit ball Bn in ℂn, i.e.(Formula presented.) if and only if there exists an A-normalized univalent...

In this paper we obtain certain sufficient conditions for the univalence of pluriharmonic mappings defined in the unit ball \(\mathbb{B}^n \) of ℂ n . The results are generalizations of conditions of Chuaqui and Hernández that relate the univalence of planar harmonic mappings with linearly connected domains, and show how such domains can play a rol...

We obtain various results related both to extreme points and to support points for the compact family (Formula presented.), where S⁰g(Bⁿ) is the family of normalized biholomorphic mappings which have g-parametric representation on the unit ball in ℂⁿ, and g is a univalent function on the unit disc U with g(0) = 1 and which satisfies certain natural...

Let f be a normalized biholomorphic mapping on the Euclidean unit ball B n in n and let α ∈ 0,1. In this paper, we will show that if f is strongly starlike of order α in the sense of Liczberski and Starkov, then it is also strongly starlike of order α in the sense of Kohr and Liczberski. We also give an example which shows that the converse of the...

We introduce a family of natural normalized Loewner chains in the unit ball, which we call ``ger\"aumig''---spacious---which allow 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...

Let X be a reflexive complex Banach space with the unit ball B. In the first part of the paper, we survey various growth and coefficient bounds for mappings in the Carathéodory family ℳ, which plays a key role in the study of the generalized Loewner differential equation. Then we consider recent results in the theory of Loewner chains and the gener...

In this paper we consider the notion of asymptotic spirallikeness in reflexive complex Banach spaces $X$ , and the connection with univalent subordination chains. Poreda initially introduced the notion of asymptotic starlikeness to characterize biholomorphic mappings on the unit polydisc in $\mathbb{C }^{n}$ which have parametric representation...

Let B be a homogeneous unit ball in X = C-n. In this paper, we obtain growth and distortion theorems for linearly invariant families F of locally biholomorphic mappings on the unit ball B with finite norm-order parallel to ord parallel to F-e,F-1. We use the Euclidean norm for the target space instead of the norm of X, because we are able to obtain...

Known results concerning the extension of normalized Loewner chains defined on the unit disk or the euclidean unit ball to higher dimensions, using either a modified Roper-Suffridge extension operator or the Pfaltzgraff-Suffridge extension operator, are shown to hold true in the more general case of L d -Loewner chains. Associated to each L d -Loew...

Let be a harmonic mapping on the unit disc in . We give some condition for to be a quasiconformal homeomorphism on and to have a quasiconformal extension to the whole plane . We also obtain quasiconformal extension results for starlike harmonic mappings of order .

In this paper we are concerned with solutions, in particular with the univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball B in ${\mathbb{C}^n}$ . We also give applications to univalence conditions and quasiconformal extensions to ${\mathbb{C}^n}$ of holomorphic...

In this paper we consider univalent subordination chains in reflexive complex Banach spaces, allowing the chains to be normalized in terms of a positive linear operator. Related adaptations in the generalized Loewner differential equation and in the notion of parametric representation are also considered. The results in this paper are generalizatio...

In this paper, we survey some recent results related to extreme points, support points and reachable families of holomorphic mappings generated by the Loewner differential equation on the unit ball Bn in ℂn. Certain applications and some conjectures are also considered.

We give a distortion theorem for linearly invariant families on the unit ball BB of a finite dimensional JB∗-triple XX by using the trace-order. The exponents in the distortion bounds depend on the Bergman metric at 0. Further, we introduce a new definition for the trace-order of a linearly invariant family on BB, based on a Jacobian argument. We a...

Let X be a complex Banach space and Y be a JB*‐triple. Let G be a bounded balanced domain in X and B Y be the unit ball in Y. Let f : G → B Y be a holomorphic mapping. In this paper, we obtain some generalization of Bohr's theorem that if a = f(0), then we have ∑k=0∞∥Dφa(a)[Dkf(0)(zk)]∥/(k!∥Dφa(a)∥)1 for z ∈ (1 / 3)G, where φ a ∈ Aut(B Y ) such tha...

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball B n in ℂn . We consider the class S 0(B n ) of biholomorphic mappings on B n which have parametric representation, i.e., they are the initial elements f(·, 0) of a Loewner chain f(z, t) = et z + … such t...

Since the work of Roper and Suffridge in 1995, there has been considerable interest in constructing holomorphic mappings of the unit ball in CnCn with various geometric properties by using lower dimensional mappings with similar properties. Such properties include convexity, starlikeness, and spirallikeness. It is also of interest to extend subordi...

We study the notion of linear invariance on the unit ball of a ⁎JBJB⁎-triple X, and we obtain some connection between the norm-order of a linear invariant family and the starlikeness of order 1/2. Also, we give some result concerning the radius of univalence of some linear invariant families. Finally, if the dimension of X is finite and if the norm...

Two-point distortion theorems are obtained for affine and linearly invariant families of harmonic mappings on the unit disk, with generalizations to pluriharmonic mappings of the unit ball in ℂ n . In particular, necessary and sufficient conditions are given for a locally univalent harmonic or pluriharmonic mapping to be univalent. Some particular...

In this paper we are concerned with solutions, in particular with univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball B n in \({\mathbb{C}^n}\). The main result is a generalization to higher dimensions of a well known result due to Becker. Various particular cases...

We determine the form of polynomially bounded solutions to the Loewner differential equation that is satisfied by univalent subordination chains of the form f (z, t)= e(tA)z + ...., where A epsilon L(C(n),C(n)) has the property m(A) > 0. Here m(A) = min{H < A(z), z >: parallel to z parallel to=1}. We also give sufficient conditions for g(z, t) = L(...

Let B be the open unit ball of a complex Banach space X and let B be homogeneous. We prove distortion results for normalized convex mappings f:B→X which generalize various finite dimensional distortion theorems and improve some infinite dimensional ones. In particular, our results are valid for the open unit balls of complex Hilbert spaces and the...

We generalize a one-variable result of J. Becker to several complex variables. We determine the form of arbitrary solutions of the Loewner differential equation that is satisfied by univalent subordination chains of the form f(z, t)=etAz+¼,{f(z, t)=e^{tA}z+\cdots,} where A Î L(\mathbbCn, \mathbbCn){A\in L(\mathbb{C}^n, \mathbb{C}^n)} has the prop...

In this paper we continue the work related to convex subordination chains in C and Cn, and prove that if f(z)=z+∑k=2∞Ak(zk) is a holomorphic mapping on the Euclidean unit ball Bn in Cn such that ∑k=2∞k2‖Ak‖⩽1, a:[0,1]→[0,∞) is a function of class C2 on (0,1) and continuous on [0,1], such that a(1)=0, a(t)>0, ta′(t)>−1/2 for t∈(0,1), and if a(⋅) sat...

Let B n be the Euclidean unit ball in C n . In this paper, we study several properties of strongly starlike mappings of order α (0<α<1) and bounded convex mappings on B n . We prove that K-quasiregular strongly starlike mappings of order α on B n have continuous and univalent extensions to [`(B)]n{\overline{B}^n}. We show that bounded convex ma...

We present a new geometric construction of Loewner chains in one and several complex variables which holds on a complete hyperbolic complex manifold M and prove that there is essentially a one-to-one correspondence between evolution families of order d and Loewner chains of the same order. As a consequence we obtain a solution for any Loewner-Kufar...

In this paper we consider non-normalized univalent subordination chains f(z, t) = exp(∫t0 A(τ)dτ )z + ... and we present the connection with the notion of generalized A-asymptotic spirallikeness on the Euclidean unit ball Bn in ℂn, where A : [0, ∞) → L(ℂn;ℂ, n) is a measurable operator that satisfies certain natural conditions.

Let X, Y be complex Banach spaces. Let G be a bounded balanced domain in X and B Y be the unit ball in Y. Assume that B Y is homogeneous. Let f: G → B Y be a holomorphic mapping. In this paper, we show that, if P = f(0), then we have Σ k=0∞ ‖ D φP (P)[D k f(0)(z k )]‖/(k!‖D φP (P)‖) < 1 for z ∈ (1/3)G, where φP ∈ AutB Y ) such that φP (P)...

In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in C n. We also obtain a sufficient condition for injectivity o...

We give starlike criteria for a class of rational mappings on the open unit ball of a complex Banach space. We also give a sufficient condition for these mappings to be convex when they are defined in Hilbert spaces. These criteria facilitate the construction of concrete examples of starlike and convex mappings on infinite dimensional domains (© 20...

In this paper we consider the notion of asymptotic starlikeness in the Euclidean space C". In the case of the maximum norm, asymptotic starlikeness was introduced by Poreda. We have modified his definition slightly, adding a boundedness condition. We prove that the notion of parametric representation which arises in Loewner theory can be characteri...

In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space ℂ n . We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric representation, where A ∈ L(ℂ n , ℂ n ), and prove tha...

Let B be the unit ball in a complex Banach space. Let S k+1*(B) be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k+1 of f(z)-z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S k+1*(B).

In this paper we consider the notion of asymptotic starlikeness in the Euclidean space C n . In the case of the maximum norm, asymptotic star-likeness was introduced by Poreda. We have modified his definition slightly, adding a boundedness condition. We prove that the notion of parametric rep-resentation which arises in Loewner theory can be charac...

We consider non-normalized univalent subordination chains and the connection with the Loewner differential equation on the unit ball in ℂ n . To this end, we study the most general form of the initial value problem for the transition mapping, and prove the existence and uniqueness of solutions. Also, we introduce the notion of generalized spirallik...