Glenn Schober

Univalent harmonic functions

Several families of complex-valued, univalent, harmonic functions are studied from the point of view of geometric function theory. One class consists of mappings of a simply-connected domain onto an infinite horizontal strip with a normalization at the origin. Extreme points and support points are determined, as well as sharp estimates for Fourier coefficients and distortion theorems. Next, mappings in Izi > 1 are considered that leave infinity fixed. Some coefficient estimates, distortion theorems, and covering properties are obtained. For such mappings with real boundary values, many extremal problems are solved explicitly.

Richardson's iteration for nonsymmetric matrices

Abstract To solve the linear N × N system (1) Ax = a for any nonsingular matrix A , Richardsons iteration (2) x j +1 = x j -α j ( Ax j - a ), j =1,2,…, n , which is applied in a cyclic manner with cycle length n is investigated, where the α j are free parameters. The objective is to minimize the error | x n +1 - x |, where x is the solution of (1). If the spectrum of A is known to lie in a compact set S , one is led to the Chebyshev-type approximation problem (3) min p -1∈ V n max z ∈ S | p ( z )|, where V n is the linear span of z , z 2 ,…, z n . If p solves (3), then the reciprocals of the zeros of p are optimal iteration parameters α j . It is shown that for a real problem (1) the iteration (2) can be carried out with real arithmetic alone, even when there are complex α j . The stationary case n =1 is solved completely, i.e., for all compact sets S the problem (3) is solved explicitly. As a consequence, the converging stationary iteration processes can be characterized. For arbitrary closed disks S the problem (3) can be solved for all n ∈ N , and a simple proof is provided. The lemniscates associated with S are introduced. They appear as an important tool for studying the stability of the iteration (2).

Curvature Estimates for some Minimal Surfaces

Let D be a simply-connected domain in the complex plane ℂ, and let f = u + iv be a univalent, orientation-preserving, harmonic mapping from D into ℂ. Then f can be written in the form f = h + \(\bar g\) where h and g belong to the linear space H(D) of analytic functions on D. In addition, f can be viewed as a solution of the elliptic partial differential equation

On schlicht mappings to domains convex in one direction

{\bar f_{\bar z}} = a{f_z}

Planar harmonic mappings

(1.1) where the function a = g′/h′ belongs to H(D) and satisfies |a(z)| < 1 for all z ∈ D. Hence the mapping f is locally quasiconformal. Conversely, any univalent solution of (1.1) with a analytic and |a| < 1 is an orientation-preserving harmonic mapping of D (see [4]). Observe that if φ is a conformal mapping from D 1 onto D, then f 1 = f o φ is a harmonic mapping defined on D 1 with D 1 with a 1 = a o φ.

Interpolation, continuation, and quadratic inequalities.

By analogy to the class of close-to-convex functions we define a class of analytic functions which are close to a family 2 of mappings onto domains convex in one direction. In contrast to the close-to-convex class the close-to-2 functions are not necessarily univalent. However, we determine the radius of convexity for 2, and this gives a lower bound for the radius of univalence of close-to-2 functions. We next derive the coefficient estimate I A.1 5nl All for close-to-2 functions and conclude with an elementary distortion theorem. 1. Mappings convex in one direction. Let 2 be the family of nonconstant analytic functionsf on the unit disk U = {I z j < 1 } satisfying the condition (1) Re{(1 _ Z2)f(z)} _ 0, z E U. If fE2, it is known [4] thatf is univalent andf(U) is a domain convex in the v-direction, i.e., the intersection of f(U) with each vertical line is connected (or empty). Moreover, f possesses the normalization lim sup Ref(z) = sup Ref(z), 2 1 IzI <1 (2) lim inf Re f(z) = inf Re f(z), z--1 ZIn <1 meaning that the prime ends corresponding to z = + 1 are, in some sense, the right and left extremes of f(U). In fact, the univalence, convexity in the v-direction, and normalization (2) characterize the class 2 [4, Theorem 1 J. We now determine the radius of convexity for the class 2. THEOREM 1. Iff EC2, then f maps {j zI <r } onto a convex domain for r < c-= 2 (1 + A/5)[ (1 + V/5) ]1/2 = .346 * * * . The constant c is sharp for the function Received by the editors July 13, 1970. AMS 1970 subject dassifications. Primary 30A32; Secondary 30A34.