A GENERALISATION OF THE CLUNIE–SHEIL-SMALL THEOREM

Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25] gave a simple and useful univalence criterion for harmonic functions, usually called the shear construction. However, the application of this theorem is limited to planar harmonic mappings that are convex in the horizontal direction. In this paper, a natural generalisation of the shear construction is given. More precisely, our results are obtained under the hypothesis that the image of a harmonic function is a union of two sets that are convex in the horizontal direction.

General univalence criteria and quasiconformal extensions starting from Loewner chains theory

The aim of this paper is to obtain general univalence conditions and quasiconformal extensions to Cn of holomorphic mappings defined on the Euclidian unit ball B. The asymptotical case of the quasiconformal extension results is also presented. We extend several results obtained by Hamada and Kohr(2011) in [15] to a more general case. In particular our results improve certain univalence criteria and quasiconformal extension results previously obtained by Pfaltzgraff [21], [22], Curt and Pascu [8], Curt [5], Hamada and Kohr [16], Curt and Kohr [6], [7] and R?ducanu [24]. As applications we present general forms of the n-dimensional version of the well-known univalence criterion due to Lewandowski [19] and its quasiconformal extension.

An extension of a basic univalence criterion

Some sufficient conditions for univalence and quasiconformal extension of a class of functions defined by an integral operator are discussed with some examples. This condition involves two arbitrary functions $ g $ and $ h $ analytic in the unit disk. A number of well-known univalent conditions would follow upon specializing the functions and the parameters involved in our main result.

REMARKS ON THE UNIVALENCE CRITERION OF PASCU AND PASCU

We consider a recent work of Pascu and Pascu [‘Neighbourhoods of univalent functions’,Bull. Aust. Math. Soc. 83(2) (2011), 210–219] and rectify an error that appears in their work. In addition, we study certain analogous results for sense-preserving harmonic mappings in the unit disc$\vert z\vert \lt 1$. As a corollary to this result, we derive a coefficient condition for a sense-preserving harmonic mapping to be univalent in$\vert z\vert \lt 1$.

General Univalence Criterion Associated with thenth Derivative

For normalized analytic functionsf(z)withf(z)≠0for0<|z|<1, we introduce a univalence criterion defined by sharp inequality associated with thenth derivative ofz/f(z), wheren∈{3,4,5,…}.