Allen Weitsman

Univalent harmonic mappings of annuli and a conjecture of J. C. C. Nitsche

Letw=f(z) be a univalent harmonic mapping of the annulus {ρ≤|z|≤1} onto the annulus {σ≤|w|≤1}. It is shown thatσ≤1/(1+(ρ2/2)(logρ)2).

Symmetrization and the Poincare metric

In the 1930s and 1940s Polya and Szegd developed an elegant theory of symmetrization which is summarized in [PS]. In this work they derived the effects of symmetrization on domain constants such as capacity, inner radius, and principal frequency. The monotonicity in the behavior of the inner radius under symmetrization yields the principle of symmetrization for simply connected regions [Ha2; p. 84] which gives bounds on the derivative of an analytic function mapping the unit disk onto a region D in terms of the conformal mapping of U onto the symmetrized region D*, with D* simply connected. Hayman later used the ideas of Polya and Szegd in the study of analytic functions [Hal] and raised the problem [Ha3; p. 32] of determining if this principle extends to general universal covering maps. The difficulty in obtaining such an extension stems from the fact that the methods rely heavily on the simple connectivity of D*. In the present work we develop a technique which does not depend on the connectivity of the region, and thus enables us to answer Haymans question affirmatively. The proof relies on a method of symmetrization introduced by Baernstein in [Bi] and further developed in [B2]. I would like to thank Professor Baernstein for several improvements and simplifications in the proofs of this paper.

On the coefficients and means of functions omitting values

Let be regular in U = {| z | f ( z ) all lie in a domain D in the w -plane. If certain geometrical restrictions are made on D we can deduce growth conditions on the maximum modulus the means and the coefficients α n . Good bounds for M ( r , f ) have been obtained under various conditions on D .

On the Gauss Map of Minimal Graphs

We consider graphs of solutions to the minimal surface equation which are unbounded over subarcs of the domain boundary. An extensive study of such surfaces was made by Jenkins and Serrin. In this note, properties of the Gauss map are studied.

Some Estimates for the Symmetrized First Eigenfunction of the Laplacian

In this paper a method is developed to study the first eigenfunction u>0 of the Laplacian. It is based on a study of the distribution function for u. The distribution function satisfies an integro–differential inequality, and by introducing a maximal solution Z of the corresponding equation, bounds obtained for Z are then used to estimate u. These bounds come from a detailed study of Z, especially the basic identity derived in Theorem 3.1.

Bounds for capacities in terms of asymmetry

The inequality (1.3) was conjectured by L. E. Fraenkel and, as noted in [6], the exponent 2 in (1.3) is sharp. The proof in [7] relies on an inequality between capacity and moment of inertia which had been proved by Pólya and Szegö [10 ; p 126] for connected sets. For general sets, this inequality had remained open until Hansen and Nadirashvili’s ingenious proof in [7]. They also showed that, in (1.3), K1 ≥ 1/4. The proofs in [6] are based on estimates for condensers.

Level sets of infinite length

The geometry of level sets plays an important role in the analysis of various function-theoretic problems. Often, however, the level sets are so complicated that one must either choose sets associated with special levels (see [3], [4]) or resort to the approximation of level sets by shorter curves (see[l, pp. 550-553]). In [3] and [4], there are some weak estimates on the length e(r, R) of the sets {z: l f (z ) I = R, Izt < r} associated with a function [ meromorphic in the plane. For such an / we do not know whether the quantity ca(r, R) can be bounded in terms of Nevanlinnas characteristic function T without reference to exceptional levels. But the following is implicit in the results in [3, pp. 121-123] and [4, p. 44]: I / [ is meromorphic in lzl-<2r and f (0 )= 1, then each subinterval [a,/3] of (O, oo) contains a set I of measure ([3a)/2 such that the inequality

Asymptotic behavior of meromorphic functions with extremal deficiencies

WITH EXTREMAL DEFICIENCIES BY ALBERT EDREI AND ALLEN WEITSMAN Communicated by Maurice Heins, September 18, 1967 Let ƒ(z) be a meromorphic function; it is assumed that the reader is familiar with the following symbols of frequent use in Nevanlinnas theory n(r,f), N(r,f), T(r,f), ô(r,f). The lower order /* and the order X of ƒ(z) are defined by the familiar relations . log T(r,f) log T(r, f) lim ml = /x, hm sup = A. r->eo l o g T r-+«> l o g T In addition to these classical concepts, we consider the total deficiency A(jf) of the function ƒ A(/)-S«(r,/) r where the summation is to be extended to all the values r, finite or oo, such that (1) 5(r, f) > 0. The number of deficient values of/, that is the number of distinct values of r for which (1) holds, will be denoted by *>(ƒ) ( ^ + <*>). The investigation presented here leads to the proof of THEOREM A. Let f(z) be a meromorphic function of lower order /x: (2) h < M < 1, and let the poles of f(z) have maximum deficiency (S(oo, ƒ) = 1). Then (3) A(/) g 2 sin TTMMoreovert if equality holds in (3), then (4) v(f) 2.

On the Boundary Behaviour of Univalent Harmonic Mappings onto Convex Domains

In this note we discuss the boundary behavior of a univalent harmonic mapping f from the unit disk U “onto” a bounded convex domain Ω in the sense of Hengartner and Schober, whose second dilatation function a is an inner function. This problem was raised by Laugesen in [10].

Uniqueness of harmonic mappings with Blaschke dilatations

AbstractLet Ω be a bounded convex domain and let ω be a finite Blaschke product of order N = 1, 2, .... It is known that the elliptic differential equation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGMbWaaSbaaSqaaiqadQhagaqeaaqabaaaaOGaai4laiaadAgadaWg% aaWcbaGaamOEaaqabaaaaa!3AFF!