摘要:

In close relation to fractional integration a one-parameter family of operators acting on a class of analytic functions regular in the unit disk is observed. In considering functionals referring to the real part, the author derives distortion inequalities showing how their bounds change after application of operators. The case of a particular family generated by a special measure is discussed in detail.

ISSN:0278-1077    

DOI:10.1080/17476938608814191

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

The goal of the present paper is to obtain new, more general sufficient conditions for univalency and quasiconformal extendibility of analytic functions in the domainby means of the general nonlinear differential operators of the form, whereFis an analytic function of its arguments (n ≥ 2). It is connected with the solution of the question about the existence of univalent solutions of nonlinear differential equations with analytic coefficients. The most natural conditions are obtained in the case of the operatorsFis nonlinear. In the paper the operators of such kind in simply connected domains with quasiconformal boundary are considered, in the case of the disk the operator with the linear part a1f(n)/f′ + a2f(n)(n= 2, 3) is also regarded.

ISSN:0278-1077    

DOI:10.1080/17476938608814192

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

This paper is a continuation of [3] where a parametric representation of Charzyński polynomials, i.e. polynomials being inverses of functions of the familiar classShas been given. By means of this parametric representation sharp bounds for coefficients as well as a sharp lower estimate of zeros of the derivative for Charzyński polynomials of fourth degree have been established.

ISSN:0278-1077    

DOI:10.1080/17476938608814193

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

LetSbe the class of functionsf(z) =z+c2z2+ … analytic and univalent in the diskLetbe forf(z) ∈S. One considers means of the formThe choicexkwhich realizes extremal properties in the classSof the functionis of main interest, especially the casexk=n-k+ 1 (k= 1,…,), which corresponds to Milin's conjecture:In 1984 L. de Branges proved this conjecture. It is known that the famous Bieberbach conjecture, Robertson's conjecture and some other conjectures for the coefficients of univalent functions hold ifInis nonpositive for alln. In this paper we use the properties ofInto obtain some inequalities for the functions of classS. In these inequalities the equality is attained by the functionKx(z).

ISSN:0278-1077    

DOI:10.1080/17476938608814194

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

The inequalitywas proved for a real parameter λ, 0 < λ < 1, by Fekete and Szegö with the Loewner method, and by J. A. Jenkins (Analytic Functions, Princeton, 1960) using the general coefficient theorem. Following his method it is proved that (1°) the above inequality holds for complex λ such thatand (2°) this inequality is sharp if and only ifwhere

ISSN:0278-1077    

DOI:10.1080/17476938608814195

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

Letfbe analytic and univalent inwith. We are interested in characterizing the functions for which (*)is bounded in any Stolz angle at. Necessary and sufficient conditions on are given in terms of extremal length as well as—for some special classes of domains—in terms of purely geometric (Euclidean) properties of . Equivalence of the boundedness of (*) and ofin any Stolz angle atis shown.

ISSN:0278-1077    

DOI:10.1080/17476938608814196

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

The paper explores some consequences of the a.e. approximability of measurable functions ofz,[zbar], by polynomials inzto the approximability of arbitrary quasiconformal mappings of a region G by Teichmüller mappings generated by analytic quadratic differentials ϕ(z). In particular, iffis a quasiconformal mapping of the unit diskD, thenfcan be approximated uniformly in Đ by such Teichmüller maps; moreover, there exists a so-calledgoodapproximation offby generalized Teichmüller maps, uniformly in Đ. On the other hand, it turns out to be possible for a sequence of Teichmüller mapsfn, to give even a good approximation of a Teichmüller mapfwithout having the corresponding quadratic differentials ϕnconverging uniformly in any neighborhood. Some of the preceding facts have their counterparts for mappings close to the identity, since the supremum ofover analytic ϕ does not differ from the supremum over measurable ϕ.

ISSN:0278-1077    

DOI:10.1080/17476938608814197

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

Letbe a simply-connected bounded domain. In [1] D. Gaier introduced the space ∑(G) of the conformal automorphisms ofGwith the topology of uniform convergence onG. In this paper it is shown that ∑(G) is locally totally disconnected ifGhas at least three prime ends not of the first kind. This implies that ∑(G) is path connected if and only ifGhas only prime ends of the first kind. Furthermore, for suchGhaving not too simple and not too wild boundary structure it is shown that ∑(G) is extremely disconnected (each element is isolated).

ISSN:0278-1077    

DOI:10.1080/17476938608814198

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

For the class of functions referred to in the title, this article finds the Koebe disk, the radius of convexity, and sharp estimates for the coefficient functionalfortin a certain interval.

ISSN:0278-1077    

DOI:10.1080/17476938608814199

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor

摘要:

We consider coefficient problems for functions in the classSof functions that are univalent in the unit disk and with the additional requirement that the derivative also be univalent in the disk. We restrict the problem further to consider only functions with real coefficients. We solve the problem of finding the maximum value of each of the even coefficientsa2a4.…a14in this restricted class. The extremal function is the same for each of these coefficients and it is given explicitly. The same function maximizes these coefficients within the larger class of typically real functions with typically real first derivative. This function definitely does not maximize the odd coefficients nor the even coefficientsa16a18.… within this larger class. The method of proof is to use sequences of polynomials from a certain restricted class of polynomials and to solve a related problem for the polynomials. These results can then be interpreted to solve the stated problem.

ISSN:0278-1077    

DOI:10.1080/17476938608814200

出版商:Gordon and Breach Science Publishers    

年代:1986

数据来源: Taylor