Sufficient conditions for the univalence of quasiconformal mappings

Using two methods, quasiconformal continuation involving a theorem of Hadamard and direct estimation of f(z₂)?f(z₁), we obtain sufficient conditions for the univalence of continuously differentiable mappings f(z) of plane domains which, in the case of conformal mappings, reduce to both well-known and new results.     

Grunsky inequalities and quasiconformal extension

The Grunsky coefficient inequalities play a crucial role in various problems and are intrinsically connected with the integrable holomorphic quadratic differentials having only zeros of even order. For the functions with quasi-conformal extensions, the Grunsky constant ℵ(f) and the extremal dilatationk(f) are related by ℵ(f)≤k(f). In 1985, Jürgen Moser conjectured that any univalent functionf(z)=z+b ₀+b ₁ z ⁻¹+… on Δ*={|z|>1} can be approximated locally uniformly by functions with ℵ(f)<k(f). In this paper, we prove a theorem confirming Moser’s conjecture, which sheds new light on the features of Grunsky coefficients. In memory of Jürgen Moser The research was supported by the RiP program of the Volkswagen-Stiftung in the Mathematisches Forschungsinstitut Oberwolfach.     

Uniform,Sobolev extension and quasiconformal circle domains

This paper contributes to the theory of uniform domains and Sobolev extension domains. We present new features of these domains and exhibit numerous relations among them. We examine two types of Sobolev extension domains, demonstrate their equivalence for bounded domains and generalize known sufficient geometric conditions for them. We observe that in the plane essentially all of these domains possess the trait that there is a quasiconformal self-homeomorphism of the extended plane which maps a given domain conformally onto a circle domain. We establish a geometric condition enjoyed by these plane domains which characterizes them among all quasicircle domains having no large and no small boundary components.     

On some results for a class of meromorphic functions having quasiconformal extension

We consider the class \(\Sigma (p)\) of univalent meromorphic functions f on \({\mathbb D}\) having a simple pole at \(z=p\in [0,1)\) with residue 1. Let \(\Sigma _k(p)\) be the class of functions in \(\Sigma (p)\) which have k-quasiconformal extension to the extended complex plane \({\hat{\mathbb C}}\), where \(0\le k < 1\). We first give a representation formula for functions in this class and using this formula, we derive an asymptotic estimate of the Laurent coefficients for the functions in the class \(\Sigma _k(p)\). Thereafter, we give a sufficient condition for functions in \(\Sigma (p)\) to belong to the class \(\Sigma _k(p).\) Finally, we obtain a sharp distortion result for functions in \(\Sigma (p)\) and as a consequence, we obtain a distortion estimate for functions in \(\Sigma _k(p).\)     

Radius of univalence of holomorphic functions

Several simple observations are made regarding the radius of univalence of functions holomorphic in the disk ¦z¦<1 and also regarding the radius of holomorphicity of the inverse functions.Translated from Matematicheskie Zametki, Vol. 7, No. 3, pp. 295–298, March, 1970.     

Univalence criterion and quasiconformal extension of holomorphic mappings

In this paper we are concerned with solutions, in particular with the univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball B in ${\mathbb{C}^n}$ . We also give applications to univalence conditions and quasiconformal extensions to ${\mathbb{C}^n}$ of holomorphic mappings on B. Finally we consider the asymptotical case of these results. The results in this paper are complete generalizations to higher dimensions of well known results due to Becker. They improve and extend previous sufficient conditions for univalence and quasiconformal extension to ${\mathbb{C}^n}$ of holomorphic mappings on B.     

On the univalence of functions with logharmonic laplacian

In this paper, we prove Radó’s theorem holds for functions of the form is logharmonic. We show that if F is of the form , where is logharmonic, then F is starlike iff ψ(z)=h(z)/g(z) is starlike. In addition, when , where L is logharmonic and H is harmonic, we give the sufficient conditions for F to be locally univalent.     

On some criteria for the univalence of analytic functions

Criteria for the univalence of functions f(z) which are regular in the domain ¦z¦ < 1 are obtained in the form of restrictions on the modulus of the quotient f (z) [f '(z)]^–1. Analogous results are obtained also for functions which are analytic in the interior of the unit disk except for a simple pole at infinity.Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 359–366, March, 1973.In conclusion the author wishes to thank L. A. Aksent'ev and F. G. Avkhadiev for discussion of the results and valuable remarks.     

Distortion functions for plane quasiconformal mappings

The authors study two well-known distortion functions, λ(K) andϕ _K(r), of the theory of plane quasiconformal mappings and obtain several new inequalities for them. The proofs make use of some properties of elliptic integrals. The work of the first author was supported in part by a grant from the United States National Science Foundation. The work of the first and second authors was supported in part by a grant from the Academy of Finland.     

Two sufficient conditions for the univalence of analytic functions

In this article we obtain sufficient conditions for the univalence of n-symmetric analytic functions in the region ¦¦>–1 and in the disk ¦¦<–1. We examine the question of univalent variation of functions analytic in ¦¦<–1 and mapping ¦¦=1 onto a contour with two zero angles. We give an application of these results to the fundamental converse boundary-value problems.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 331–346, March, 1976.In conclusion the author would like to thank L. A. Aksent'ev for his guidance, and those who took part in his seminar for their useful advice.     

On analytic functions related with generalized Robertson functions

Analytic functions f are called Robertson functions for which zf^′ is α-spiral-like. This concept is generalized by several authors and a class real, of analytic functions is introduced and studied. It is noted that the functions in are of bounded boundary rotation and consists of Robertson functions.In this paper, we use the class to define a new class of analytic functions which unifies a number of classes previously studied such as the class of close-to-convex functions of higher order. Some interesting properties of this class, including coefficient problems, inclusion results and a sufficient condition for univalency are studied.