Ted J. Suffridge

Norm order and geometric properties of holomorphic mappings in\(\mathbb{C}^n \)

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.

Extreme points for convex mappings ofB n

We exhibit a collection of extreme points of the family of normalized convex mappings of the open unit ball of ℂn forn≥2. These extreme points are defined in terms of the extreme points of a closed ball in the Banach space of homogeneous polynomials of degree 2 in ℂn−1, which are fully classified. Two examples are given to show that there are more convex mappings than those contained in the closed convex hull of the set of extreme points here exhibited.

Linear Invariance, Order and Convex Maps in

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.

A Reflection Principle for Proper Holomorphic Mappings of Strongly Pseudoconvex Domains and Applications

This result is proved in [5]. The techniques of [12] and [5] involve detailed computat ion of differential invariants on the boundary. Another paper, philosophically related to the above but involving more elementary techniques, is [1] in which proper holomorphic maps of B n to B, are shown to be automorphisms. More recently this result has been proved on strongly pseudoconvex domains in [2] and in [4]. In the paper [3], which is the starting point for the present work, completely elementary means are used to obtain a reflection principle for suitable maps of B,, to B k. This in turn leads to a proof of the result of [12] assuming only C 2 smoothness to the boundary. Further, detailed information is obtained about the maps even when k > n + 1. The elementary techniques of [3] may be adapted to use on strictly

A continuous extension of the de la Vallée Poussin means

AbstractAmong the many interesting results of their 1958 paper, G. Pólya and I. J. Schoenberg studied the de la Vallée Poussin means of analytic functions. These are polynomial approximations of a given analytic function on the unit disk obtained by taking Hadamard products of the functionf with certain polynomialsVn(z), wheren is the degree of the polynomial. The polynomial approximationsVn *f converge locally uniformly tof asn→∞. In this paper, we define a subordination chainVλ(z),γ>0, |z|<1, of convex mappings of the disk that for integer values is the same as the previously definedVn(z). Iff is a conformal mapping of the diskD onto a convex domain, thenVλ*f→f locally uniformly as λ→∞, and in fact

Construction of Convex Mappings of p-Balls in ℂ2

V_{\lambda _1 } * f(\mathbb{D}) \subset V_{\lambda _2 } * f(\mathbb{D}) \subset f(\mathbb{D})

Construction of close-to-convex harmonic polynomials

when λ2 > λ1. We also consider Hadamard products of theVλ with complex-valued harmonic mappings of the disk.

The inner mapping radius of harmonic mappings of the unit disk

In this paper, we introduce a method of constructing sense-preserving harmonic polynomials, The polynomials considered generally have the form where h and g are polynomials in the complex variable z witgh h(0) = g(0)=g′(0)=0 and g′(z)=B(z)h′(zi) where B is a finite Blaschke product. The results include definition of a family of typically real harmonic polynomials that have some interesting geometric properties. In addition, we discuss coefficient regions for univalent harmonic polynomials.