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Applications of a decomposition of holomorphic mappings in Cn with respect to a cyclic group
Abstract
In the paper the decomposition with respect to the group of roots of unity is applied to characterize some classes of biholomorphic mappings in There is considered the problem of subordination and majorization relations between convex mappings and their components in the above partition. There is also given a distortion theorem for convex mappings and a criterion for starlikeness in
... Continuing the first section we give a very useful functional symmetry. In the papers [3,6,11] there are considered the consequences of a modification of the above starlikeness factorization (1), using a unique decomposition of mappings f : B n → C n with respect to the cyclic group of k-th. roots of unity, k ≥ 2. Below we present such partition for mappings f : X ⊃ → Y, where X, Y are normed complex vector spaces and is a k-symmetric nonempty subset of X (ε = for the generator ε = exp 2πi k of the above group) [14]. ...
... , we obtain that in the starlikeness factorization (1) the (1, k)-symmetry of f implies the same for the mapping h ∈ P. A reverse statement to this observation is more important. An open problem in this direction can be found in the paper [11] and its solution is given in [6]. In the paper [11] there was considered a case where the map f on the right or left side of the starlikeness factorization (1) is replaced by its (1, k)-symmetrical part f 1,k . ...
... An open problem in this direction can be found in the paper [11] and its solution is given in [6]. In the paper [11] there was considered a case where the map f on the right or left side of the starlikeness factorization (1) is replaced by its (1, k)-symmetrical part f 1,k . We recall only one of the possible cases. ...
It is known that the starlikeness plays a central role in complex analysis, similarly as the convexity in functional analysis. However, if we consider the biholomorphisms between domains in C n , apart from starlikeness of domains, various symmetries are also important. This follows from the Poincaré theorem showing that the Euclidean unit ball is not biholomorphically equivalent to a polydisc in C n , n > 1 . From this reason the second author in 2003 considered some families of locally biholomorphic mappings defined in the Euclidean open unit ball using starlikeness factorization and a notion of k -fold symmetry. The 2017 paper of both authors contains some results on the absorption by a family S ( k ) , k ≥ 2 , of the above kind, the families of mappings biholomorphic starlike (convex) and vice versa. In the present paper there is given a new sufficient criterion for a locally biholomorphic mapping f , from the Euclidean ball B n into C n , to belong to the family S ( k ) , k ≥ 2 . The result is obtained using an n -dimensional version of Jack’s Lemma.
The subject of the paper concerns the existence of different inclusions between families St, Sc of C-n - biholomorphic starlike or convex mappings and a family S(k), k >= 2, of C-n - locally biholomorphic mappings. In the definition of S(k) we used a combination of a Kikuchi-Matsuno-Snifridge's starlikeness condition and a property of (j, k)-symmetrical mappings in C-n. This definition generalizes a notion of planar Sakaguchi's functions onto n-dimensional case. A motivation for the family S(k) comes from the paper [11] of the second author and the paper [6] of Hamada and Kohr. The family S(k) is a superset of a family which is connected with a problem posed in the first paper and solved in the second one.
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 B is spirallike of type α and k-fold symmetrical. When α=0, this result solves a natural problem that is similar to an open problem posed by Liczberski. We also give two examples of k-fold symmetrical Loewner chains.
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.
ⅠMore than fifty years ago, Henri Cartant ~j suggested that geometric function theory ofone complex variable should be extended to biholomorphic mappings of several complexvariables. In particular, he cited the special classes of starlike and convex mappings asappropriate topics for generalization. In noting some of the difficulties of generalization, hepointed out the Growth Theorem as one of the results that would not extend to thepolydisc (nor to the ball). Also, he observed that for normalized biholomorphic mappings
Not many convex mappings on the unit ball in C n for n > 1 are known. We introduce two families of mappings, which we believe are actually identical, that both contain the convex mappings. These families which we have named the "Quasi-Convex Mappings, Types A and B" seem to be natural generalizations of the convex mappings in the plane. It is much easier to check whether a function is in one of these classes than to check for convexity. We show that the upper and lower bounds on the growth rate of such mappings is the same as for the convex mappings.
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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.
In the recent paper [Math. Bohem. 120, No. 1, 13-28 (1995; Zbl 0838.30004)] we showed, in an elementary way, that every complex valued function can be presented, on a k-symmetrical subset of the complex plane, as a finite sum x k 0 +x k 1 +⋯+x k k-1 of (ℓ,k)-symmetrical functions. This partition is a generalization of the well-known fact, that every function is the sum of an even function and an odd function. In the present paper we extend our considerations onto the case if k tends to infinity. As a result we obtain a series with symmetrical components. This series is similar to the Laurent series, but the constant coefficients are replaced by the functions which are constant on the circles centred at the origin.
The Roper-Suridge extension operator, originally introduced in the context of convex functions, provides a way of extending a (locally) univa- lent function f 2 Hol(D;C) to a (locally) univalent map F 2 Hol(Bn;Cn). If f belongs to a class of univalent functions which satisfy a growth theorem and a distortion theorem, we show that F satises a growth theorem and conse- quently a covering theorem. We also obtain covering theorems of Bloch type: If f is convex, then the image of F (which, as shown by Roper and Suridge, is convex) contains a ball of radius =4. If f 2 S, the image of F contains a ball of radius 1=2.
Let n be an arbitrary positive integer, We decompose the Laguerre polynomials Lm(α) as the sum of n polynomials Lm(α,n,k); m ∈ N; k = 0, 1,…, n − 1; defined by Lm(x,n,k)(z)=1n∑t=0n−1exp−2iτklnLm(x)z exp2iτln,zϵCIn this paper, we establish the close relation between these components and the Brafman polynomials. The use of a technique described in an earlier work [2] leads us firstly to derive, from the basic identities and relations for Lm(α), other analogous for Lm(α,n,k) that turn out to be two integral representations, an operational representation, some generating functions defined by means of the generalized hyperbolic functions of order n and the hyper-Bessel functions, some finite sums including multiplication and addition formulas, a non standard (2n + 1)-term recurrence relation and a differential equation of order 2n. Secondly, to express some identities of Lm(α) as functions of the polynomials Lm(α,n,k). Some particular properties of Lm(α,n,0), the first component, will be pointed out.
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}.
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.
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.
The coefficient bounds and the growth and distortion theorems for convex functions in one complex variable are generalized to several variables. The holomorphic mappings studied are defined in the unit ball or some other domain of one of the first three classical types. Each mapping takes its domain onto a convex set in a one-to-one fashion. The coordinate functions of each mapping have multivariable power series about the origin. The best possible upper bounds are found for certain combinations of the coefficients of these power series. In case the domain is the unit disk in the plane, these bounds reduce to the classical coefficient estimates for convex functions. As an application, these coefficient bounds are used to obtain the best possible upper and lower bounds on the growth of the magnitude of each mapping in terms of the magnitude of the independent variable. Also, estimates on the magnitudes of various derivatives of each mapping are found.
Thesis (Ph. D.))--University of Kentucky, 1995. Abstract ([1] leaf) bound with copy. Vita. Includes bibliographical references (leaf 76).
Introduction. In the present paper the authors study some families of functions from a complex linear space X into a complex linear space Y . They introduce the notion of (j, k)-symmetrical function (k = 2, 3, . . .; j = 0, 1, . . . , k - 1) which is a generalization of the notions of even, odd and k-symmetrical functions. It has turned out that for every function x defined on a k-symmetrical subset U of X there exists exactly one sequence (y 0 , y 1 , . . . , y kGamma1 ) of (j, k)-symmetrical functions y j such that x = y 0 + y 1 + . . . + y<F8.4
Some remarks on the subordination of holomorphic mappings from the unit ball in Cn into Cn
- Liczberski
P. Liczberski, Some remarks on the subordination of holomorphic mappings from the unit ball in C n into C n, Sci. Bull. Łód´. Univ. Math. 27 (1991) 31–42.
On the application of (j,k)-symmetrical functions to solving some functional equations
- Liczberski