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Distortion theorems for biholomorphic convex mappings in C n
Abstract
In the paper the problem of sharp lower estimation for ‖Df(z)‖ in the class of normalized biholomorphic mappings f between the open unit ball Bn and convex domains in Cn has been considered.
... On the other hand, for f ∈ K, Liczberski-Starkov [8] gave a sharp lower bound for DΦ n (f )(z) near the origin. They gave a conjecture that the sharp lower bound holds on B n . ...
... Let z, w = n j =1 z j w j be the Euclidean inner product and z = z, z 1/2 be the Euclidean norm. Liczberski-Starkov [8] showed that the equality ...
... Remark 7. For f ∈ K, we have the following sharp upper bound [9], [8]: ...
Liczberski–Starkov gave a sharp lower bound for ‖DΦn(f)(z)‖ near the origin, where Φn is the Roper–Suffridge extension operator and f is a normalized convex mapping on the unit disk in C. They gave a conjecture that the sharp lower bound holds on the Euclidean unit ball Bn in Cn. In this paper, we will give a sharp lower bound on Bn for a more general extension operator and for normalized univalent mappings f or normalized convex mappings f. We will give a lower bound for mappings f in a linear invariant family. We will also give a similar sharp lower bound on bounded convex complete Reinhardt domains in Cn.
... In [10], it is proved that the upper bound in Theorem 1.1(i) is sharp for the open unit balls of complex Hilbert spaces although the lower bound is not sharp for the Euclidean balls of dimension at least 2 (cf. [18]). Zhu and Liu [21] have obtained the following distortion theorem for convex mappings f on the open unit balls of complex Banach spaces: ...
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 Cartan domains.
The invariance of strong and almost spirallike mappings of type beta and order alpha is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallikeness of type beta and order alpha on the unit ball B-n in C-n and on bounded and complete Reinhardt domains. Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirallikeness of type beta, strong and almost starlikeness of order alpha, strong starlikeness on the corresponding domains. Thus we can construct more subclasses of spirallike mappings in several complex variables.
The invariance of strong and almost spirallike mappings of type β and order a is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallikeness of type β and order α on the unit ball Bn in ℂn and on bounded and complete Reinhardt domains. Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirallikeness of type β, strong and almost starlikeness of order α, strong starlikeness on the corresponding domains. Thus we can construct more subclasses of spirallike mappings in several complex variables.
In this paper, we consider the problem of distortion theorems for mappings which map the unit ball biholomorphically onto
convex domains in ℂ
n
. In particular, we discuss two distortion conjectures for such mappings.
In this article, we give some distortion theorems for biholomorphic convex mappings in Banach spaces and Hilbert spaces. In particular, we prove that the conjecture of Hamada and Kohr is true in Banach spaces.
In this paper we give a survey about the Roper-Suffridge extension operator and the developments in the theory of univalent
mappings in several variables to which it has led. We begin with the basic geometric properties (most of which now have a
number of different proofs) and discuss relations with the theory of Loewner chains and generalizations and modifications
of the operator, some of which are very recent.
Let Jf be the Jacobian of a normalized biholomorphic mapping / from the unit ball B2 into C2 . An expression for the log det Jf is deter-mined by considering the series expansion for the renormalized mappings F obtained from / under the group of holomorphic automorphisms of B2 . This expression is used to determine a bound for |det/y| and |arg det//I for / in a compact family X of normalized biholomorphic mappings from B2 into C2 in terms of a bound C(X) of a certain combination of second-order coeffi-cients. Estimates are found for C(X) for the specific family X of normalized convex mappings from B2 into C2.
In this paper we consider univalent maps of domains in Cn(n ≧ 2). Let P be a polydisk in Cn. We find necessary and sufficient conditions that a function f:P→ Cn be univalent and map the polydisk P onto a starlike or a convex domain. We also consider maps from Dp = {z: | z|p < 1} ⊂ Cn(formula presented)into Cn and give necessary and sufficient conditions that such a map have starlike or convex image.
Let Jf be the Jacobian of a normalized biholomorphic mapping from the unit ball B2 into C2. An expression for the log det Jf is determined by considering the series expansion for the renormalized mappings F obtained from under the group of holomorphic automorphisms of B2. This expression is used to determine a bound for and arg deI for in a compact family X of normalized biholomorphic mappings from B2 into C2 in terms of a bound C(X) of a certain combination of second-order coefficients. Estimates are found for C(X) for the specific family X of normalized convex mappings from B2 into C2.
In the complex plane, univalent analytic functions which map the unit disk onto starlike domains or onto convex domains are characterised by simple analytic conditions and have been extensively studied. In higher dimensions, to demand that a mapping takes the unit ball to a convex domain turns out to be a very restrictive condition. Indeed, examples are rather hard to construct. In this paper after a brief mention of some of the well-known results in the plane, we prove a theorem which enables us to find a convex mapping in higher dimensions given a convex function in the plane. We begin with some definitions. �9 LetA={zEC:lz[< 1}.
In this paper we suggest a new definition of the order of a linearly invariant family of locally biholomorphic mappings of the unit ball in . This definition is equivalent to the one given by Pfaltzgraff in [J.A. PfaltzgraffComplex Variables Theory Appl 33 (1997), 239–253.]. It bases on a very simple relationship with the Jacobian of the mappings (see Corollary 1). It appears that the order of a mapping depends only on its Jacobian (see Proposition 1).
We continue the study, begun by the first author, of linear-invariant families in . We obtain the unexpected result that the Cayley transform (the analogue of the half-plane mapping in the complex plane) does not give the maximum or minimum distortion among all mappings of the unit ball of n≥ 2, onto convex domains. In addition a result analogous to that of Pommerenke that a linear invariant family has order 1 (the smallest possible order) if and only if it consists of convex mappings of the disk does not hold for the ball in n≥ 2. Finally, we extend these ideas to the polydisk in . The theory for this case bears some similarity to the theory for the ball but there are some striking differences as well. For example it is true that the family of convex holomorphic mappings of the polydisk in has minimum order (which is nin this case) but it is not true that for n>1 every linear-invariant family of minimum order consists of convex mappings.
We give a version of the distortion theorem for biholomorphic mappings between the unit polydisc and convex domains in the space of n complex variables.
Distortion Bounds mappings, in linear invariant families of locally hiholomorphic mappings from the n-Ball into Cn are established. Coordinate-free methods yield a short and straightforward proof of the distortion theorem. The n-dimensional versions of linear-invariant families, the order of a family and the Koebe transform are introduced. A method for generating many examples of linear-invariant families is presented.
We introduce a new notion of the order of a linear invariant family of locally biholomorphic mappings on then-ball. This order, which we call the norm order, is defined in terms of the norm rather than the trace of the “second Taylor
coefficient operator” of mappings in a family. Sharp bounds on ‖Df(z)‖ and ‖f(z)‖, a general covering theorem for arbitrary LIFs and results about convexity, starlikeness, injectivity and other geometric
properties of mappings given in terms of the norm order illustrate the useful nature of this notion. The norm order has a
much broader range of influence on the geometric properties of mappings than does the “trace” order that the present authors
and many others have used in recent years.
The Roper-Suffridge extension operator provides a way of extending a (locally) univalent functionfεH(U) to a (locally) biholomorphic mappingF∈H(Bn). In this paper, we give a simplified proof of the Roper-Suffridge theorem: iff is convex, then so isF. We also show that iff∈S
*, theF is starlike and that iff is a Bloch function inU, thenF is a Bloch mapping onB
n. Finally, we investigate some open problems.
The notion of a linearly invariant family of mappings of a ball in C was introduced in the article ``Pfaltzgraff J. A., Distortion of locally biholomorphic maps of the n-ball, Complex Variables, 33, 239-253 (1997)''''; it generalizes the classical case n=1 studied earlier by Ch. Pommerenke and other authors. In the indicated article, Pfaltzgraff in particular obtained and used a false equality (5.3). Application of this equality also underlies some assertions in other articles. Consequently, some theorems remain unproven. We propose the dimension reduction method which enables us to save the proof and obtain new results on linearly invariant families of mappings of a ball. The idea of the method is simple and consists in reduction of a problem posed for linearly invariant families in C
n to a problem for the classical case of a disk (n=1).
In this paper, using the Bergman kernel function KD(z, z), we give necessary and sufficient conditions that a pseudoconformal mapping f(z) be starlike or convex in some boundedschlicht domain D for which the kernel function KD(z, z) becomesinfinitely large when the point z € D approaches the boundaryof D in any way. We also consider starlike and convex mappings from the polydisk or unit hypersphere into Cn.
Geometric properties of some classes of holomorphic mappings in Cn
- Liczberski