摘要:

The boundary condition Re(Pw) = ϕ of an elliptic system in a bounded domain of the complex plane can be transformed into the Dirichlet boundary condition Re ω = ϕ (by means of the reversible substitution ω= Pw) if it is possible to extend the matrix-valued functionPcontinuously to the whole of Ǥ without violating the Lopatinski condition detP≠ 0. Due to W. L. Wendland [8] this problem was solved by W. Tutschke [7] under special assumptions. Also in connection with elliptic systems W. L. Wendland in [8] used a homotopy fromPto the unit matrix, constructed by piecewise Gauss elimination, but this method can fail as was shown by a counter-example. A complete but rather deep-lying result is contained in Bott's periodicity theorem, stating that with the functionsPa- valued functional GradPcan be associated and the above mentioned problems are solvable if one has GradP= 0. It is then easy to prove Bojarski's theorem [1], that GradPis the number of revolutions of the vector detPduring one circulation around the boundary. GradPis an important and well-known quantity in the theory of elliptic systems in the plane. In the present paper we give an elementary proof of the above mentioned special part of Bott's theorem by constructing a homotopy fromPto the unit matrix in the case GradP=0.

ISSN:0278-1077    

DOI:10.1080/17476938608814181

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

We consider the following problem. Letfbe a function regular and of exponential type τ in the half planewhich satisfiesfor n = 0,1,2,… andj= 1,…, p, wherekjare integers with 0 < … <kpExponential type τ means thatfsatisfies the inequalityfor each positive ε. The problem is to find the greatest τ that allows to prove thatfis identically zero or a certain special function. Special cases of this problem were solved by Carlson [3], Korevaar [5] and Gervais and Rahman [4]. We extend and generalize their results and give a complete solution of the problem ifkp≤4. In [2] we have already considered this problem for entire functions, wherefand the derivativesvanish at all integers.

ISSN:0278-1077    

DOI:10.1080/17476938608814182

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

A classical result of Bergman and Schiffer, on analytic continuation of functions of two complex variables in terms of quadratic inequalities, is generalized to an abstract setting. The resuhs in this setting contain previous results by FitzGerald, Donoghue, Hengartner and Schober, and Hamilton.

ISSN:0278-1077    

DOI:10.1080/17476938608814183

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

Herrn Professor Helmut Grunsky gewidmet In an earlier joint paper of the first author, the following theorem was “Supposefis a nonconstant meromorphic function in the complex plane, that shares two finite nonzero values counting multiplicities with one of its derivatives. Thenfis identical with that derivative.” In this paper we settle the case that one of the shared values is allowed to be zero. The claim is then the same.

ISSN:0278-1077    

DOI:10.1080/17476938608814184

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

Clausen's formula for the product of two hypergeometric series and a summation formula are used to give a short proof of an inequality used by de Branges in his proof of the Bieberbach, Robertson, and Milin conjectures. Some possible extensions of the de Branges-Milin inequality are pointed out.

ISSN:0278-1077    

DOI:10.1080/17476938608814185

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

LetSbe the usual class of normalized univalent functions. We write

ISSN:0278-1077    

DOI:10.1080/17476938608814186

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

For Helmut Grunsky in affection and admiration

ISSN:0278-1077    

DOI:10.1080/17476938608814187

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

Letbe a given bijective analytic mapping of the circumferenceCof the unit circle onto itself. By the uniformization theorem, there exists an analytic Jordan curve Г with the following property: there are two bijective conformal mappingsF(interior of C→ interior of Г) andG(exterior of C → exterior of Г) such thatfor all. We consider this conformal welding under the assumption thatG(∞) = ∞. Our main result is a relation (Theorem 2) which will allow—at least in certain cases—an estimate of the rotation of Г in terms of the given function Ф.

ISSN:0278-1077    

DOI:10.1080/17476938608814188

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

LetU(p)denote the family of functionsf(z) meromorphic and univalent in |z] < 1, having a simple pole atp, 0 <p< 1, and normalized byf(0) = 0f′(0) = 1. Let S(p) denote the family of functionsf(z) regular and univalent in the open unit disk slit along the segment p ⩽ z < 1 and normalized byf(0) = 0f′(0) = 1. The author obtains the minimum of |f(z)|, |z|=r,0<1, the maximum of |f(z)|, |z|=r, 0 <r<pand the maximum of |f′(z)|, |z| =r,0<r<p, forfεS(p). He also obtains the maximum of |f(z)|, |z| =rp<r<1 forfεU(p). In all cases the bound is uniquely assumed for the function z[1 - (p+p-1)z + z2]-1In addition the author obtains the exact set of values which can be omitted by some function in U{p).

ISSN:0278-1077    

DOI:10.1080/17476938608814189

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

LetPn= {pp{z) = z + a2z2+…+ anzn} and define μ(p) = min{|z|:p′(z)=0}pεPn. The paper deals with the following problem: Given ⊆Pn, how to obtain the best bound r(f) such thatpεPn, μ(p) ⩾ r(f) impliespεf. For classesfwith a certain test-property the extremal polynomials are of type [(z + r)n− rn]/(nrn-1). The main result states thatf⊆Pnhas the test-property, if the polynomials offcan be described by a criterion of Dieudonné-type. This result contains classical theorems of Alexander and Kakeya.

ISSN:0278-1077    

DOI:10.1080/17476938608814190

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor