Planar Harmonic Mappings Research Articles

Schwarzian derivative criteria for valence of analytic and harmonic mappings

AbstractFor analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass–Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally, certain classes of harmonic mappings are shown to have finite Schwarzian norm.

Connections between various subclasses of planar harmonic mappings involving hypergeometric functions

The main purpose of this paper is to establish connections between various subclasses of harmonic mappings in the plane by applying certain convolution operators involving hypergeometric functions. To be more precise, we investigate such connections with harmonic k-uniformly starlike and harmonic k-uniformly convex mappings in the plane.

Planar harmonic convolution operators generated by hypergeometric functions

Let S Ĥ be the family of all planar harmonic, univalent and sense-preserving mappings f=h+g¯ where h and g are analytic funtions in the open unit disk. The purpose of this article is to investigate connections between the theory of harmonic mappings in the plane and hypergeometric functions by applying certain planar harmonic convolution operators on various subclasses of S Ĥ . Dedicated to Professor Eveyln M. Silvia, late Professor of Mathematics, UC Davis February 8, 1948–January 21, 2006

Univalence criteria for lifts of harmonic mappings to minimal surfaces

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass-Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformai metrics and on a Schwarzian introduced by Ahlfors for curves. Convexity plays a central role.

Local Decomposition of Planar Harmonic Mappings

A necessary and sufficient condition is given for a planar harmonic mapping f to be locally decomposable as a univalent harmonic mapping F of an analytic function φ.

Landau’s Theorem for Planar Harmonic Mappings

Landau gave a lower estimate for the radius of a schlicht disk centered at the origin and contained in the image of the unit disk under a bounded holomorphic function f normalized by f(0) = f′(0) − 1 = 1. Chen, Gauthier, and Hengartner established analogous versions for bounded harmonic functions. We improve upon their estimates.

Geometric Properties of Harmonic Shears

This paper is a study of planar harmonic mappings produced with the “shear construction” devised by Clunie and Sheil-Small in 1984. Specifically it will describe the geometry of mappings produced by the shear construction. The first section introduces basic concepts, including a description of the shear construction itself, a technique for constructing examples of harmonic mappings by shearing a conformal mapping. The main body of the paper alternates between presenting examples of harmonic shears, illustrating them graphically with the help of Mathematica software, and using an integral representation of the harmonic shear to explain and predict characteristics of the image domain.

Convolutions of planar harmonic convex mappings

Ruscheweyh and Sheil-Small proved that convexity is preserved under the convolution of univalent analytic mappings in K. However, when we consider the convolution of univalent harmonic convex mappings in , this property does not hold. In fact, such convolutions may not be univalent. We establish some results concerning the convolution of univalent harmonic convex mappings provided that it is locally univalent. In particular, we show that the convolution of a right half-plane mapping in with either another right half-plane mapping or a vertical strip mapping in is convex in the direction of the real axis. Further, we give a condition under which the convolution of a vertical strip mapping in with itself will be convex in the direction of the real axis

Bloch constants for planar harmonic mappings

We give a lower estimate for the Bloch constant for planar harmonic mappings which are quasiregular and for those which are open. The latter includes the classical Bloch theorem for holomorphic functions as a special case. Also, for bounded planar harmonic mappings, we obtain results similar to a theorem of Landau on bounded holomorphic functions.

On the dilatation of univalent planar harmonic mappings

It is shown that if f f is a univalent harmonic mapping of the unit disk onto a domain having a smooth boundary arc which is convex with respect to the domain, and if the dilatation has modulus 1 on the arc, then the arc must be a line segment.

A decomposition theorem for planar harmonic mappings

A necessary and sufficient condition is found for a complex-valued harmonic function to be decomposable as an analytic function followed by a univalent harmonic mapping.

Constants for Planar Harmonic Mappings

Constants for Planar Harmonic Mappings