The Roper-Suffridge extension operator provides a way of extending a (locally) univalent functionfeH(U) to a (locally) biholomorphic mappingF∈H(Bn). In this paper, we give a simplified proof of the Roper-Suffridge theorem: iff is convex, then so isF. We also show that iff∈S
 *, theF is starlike and that iff is a Bloch function inU, thenF is a Bloch mapping onB
 n. Finally, we investigate some open problems.

Univalent mappings associated with the Roper-Suffridge extension operator

On the faber polynomials of the univalent functions of class

Universal Teichmüller space and BMO

In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there?

/pdf/complete-linear-weingarten-surfaces-of-bryant-type-a-plateau-2wibu91tbk.pdf

Complete linear weingarten surfaces of Bryant type. A plateau problem at infinity

This is a survey article on domain constants related to uniform perfectness. We gather comparison theorems for various domain constants, most of which are more or less known or elementary but not stated quantitatively in the literature, and some are new or improved results. Among these theorems, our main result is a comparison of the modulus and the injectivity radius of a hyperbolic Riemann surface. Its proof relies upon a comparison of extremal and hyperbolic lengths, which seems to be interesting in itself. And we include a lower estimate of the Hausdorff dimension of a compact set in the Riemann sphere by the modulus of its complement. We also discuss the variance of these domain constants under conformal, quasiconformal or Mobius maps.

https://ir.lib.hiroshima-u.ac.jp/files/public/1/17230/20141016130730649671/CmpVar_36_311.pdf

Various domain constants related to uniform perfectness

Five domain constants are studied in our paper, all related to the hyperbolic geometry in hyperbolic plane regions which are uniformly perfect (in Pommerenke’s terminology). Relations among these domain constants are obtained, from which bounds are derived for the variance ratio of each constant under conformal mappings of the regions, and we also show that each constant may be used to characterize uniformly perfect regions.

Quasi-invariant domain constants

Several properties of a certain series of differential operators which are invariant under the Mobius group (Aharonov invariants) are proved, and in terms of this series new conditions for univalence and quasiconformal extendability of meromorphic functions are established.

Aharonov invariants and univalent functions

Bergman kernel function and univalence criteria

In this paper we study various classes of locally convex analytic functions in the unit disc, which are invariant under the group of Mobius automorphisms of the unit disc. Bounds for the Schwarzian derivative of functions in these classes are achieved and used to obtain estimates for the uniform hyperbolic radius of univalence in these classes.

Reuven Harmelin

Papers

Quasi-invariant domain constants

Aharonov invariants and univalent functions

Bergman kernel function and univalence criteria

Locally convex functions and the Schwarzian derivative

On the Derivatives of the Schwarzian Derivative of a Univalent Function and their Symmetric Generating Function