摘要:

The second differenceis important in the study of differentiability properties of functions of a real variable. We consider a similar quantity to describe the geometry of a Jordan curveCin the plane, with me aim of relating it to the existence ot tangents toC. Here we establish conditions that guarantee thatCis a quasicircle or an asymptotically conformal curve.

ISSN:0278-1077    

DOI:10.1080/17476939308814622

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

Let Ω be a simply connected domain on the Riemann sphere, andF: D → Ω be a conformal mapping of the unit disk onto Ω. We prove integral mean inequalities of the formfor certain functionsfwhereFn(z)=F(zn)anduis subharmonic on the range off. These reverse the usual subordination inequalities. We also prove results of this type whenfandFnare polynomials, with the zeros ofFnuniformly spaced on a circle. These results may bear on some well known open extremal problems for which the extremals are expected to haven–fold symmetry.

ISSN:0278-1077    

DOI:10.1080/17476939308814623

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

We prove that iffmaps the unit diskDonto a convex domain withF(0)=0, thenwith equality holding for eachfthathas the prupeiiy that the left and right-nana tangents at each pointwε∂(D) are tangent to the circleThis verifies the conjecture which we made in the early 1980's, which arose of our work on omitted value problems for classes of univalent functions. This conjecture was announced at the Purdue Symposium on the Occasion of the Proof of the Bieberbach Conjecture in 1985 and published in the article “Open Problems and Conjectures in Complex Analysis”(Computational Methods and Function TheoryLecture Notes in Math. #1435, 1990 pp. 1-26).

ISSN:0278-1077    

DOI:10.1080/17476939308814624

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

Let ф be a simple closed convex curve ofand letf* be a homeomorphism from the circleonto ф. Let p be the mean value off*over. Then the solution of the Dirichlet problemf=f*onandonis univalent on the annulus.

ISSN:0278-1077    

DOI:10.1080/17476939308814625

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

Circle packing imbeddings determine quasiconformai mappings which, in certain situations, approximate conformal mappings. In the more general case of circle packing immersions, it has not been proven that the associated immersion mappings are again quasiregular (i.e., of bounded dilatation but not necessarily univalent). If this were true, then their muitisheeted image surfaces would be regularly exhaustible and consequently Ahlfors' value distribution theory for covering surfaces could be applied to them. In this paper, although we do not settle the question or. whetner such immersions are quasiregular, we are able to prove directly from length-area estimates that the property of being regularly exhaustible holds for the muitisheeted image surfaces. Thus, for example, a Picard type theorem holds: The image of an entire circle packing immersion omits at most one finite value in the complex plane,

ISSN:0278-1077    

DOI:10.1080/17476939308814626

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

We prove that positive solutions of the nonlinear Dirichlet problemwhere Ω is a ball or an annulus inandis a polynomial with nonnegative coefficients, have alternating power series representations at the origin. We also obtain several existence and nonuniqueness results for certain functions of the formwiththe critical exponent, whereWe give lower bounds forRin terms of the initial valueu(0), whereRis the radius of the ball Ω, an upper bound foru(r) for all 0<r<R, and an upper bound for|′(R)|. We also give several results for multiple solutions of this problem, showing that two such solutions must cross in (0,R), and give lower bounds forRin terms of the value of the crossover in this situation, as well as a nonexistence result.

ISSN:0278-1077    

DOI:10.1080/17476939308814627

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

Under normalized conformal mappings of a multiply connected domain ω it is found that the sharp lower bound for the capacity of the image of a given setis the Robin capacity ofAwith respect to ω. The result is generalized to a sharp inequality for quadratic forms involving Green's function and the Robin function associated withAand ω.

ISSN:0278-1077    

DOI:10.1080/17476939308814628

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

Assume that a compact setwhich lies inand which has capacity c is mapped by a functionfschlicht in. Let EFbe the image ofEunderF, and let cFbe the capacity of EF. We investigate the dependence of cp onRfor largeR.

ISSN:0278-1077    

DOI:10.1080/17476939308814629

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

Many universal constraints for a discrete Möbius groupGare obtained by studying the sequence

ISSN:0278-1077    

DOI:10.1080/17476939308814630

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor

摘要:

This paper studies the radial limit and me radial growth of functions defined bywhere μ is a complex-valued Borel measure. The results depend on local properties of the measure μ. Various nontangential limits associated withfare shown to hold off exceptional sets whose ß -capacity is zero.

ISSN:0278-1077    

DOI:10.1080/17476939308814631

出版商:Gordon and Breach Science Publishers    

年代:1993

数据来源: Taylor