We study some classes of planar harmonic mappings produced with the shear construction devised by Clunie and Sheil-Small in 1984. The first section reviews the basic concepts and describes the shear construction. The main body of the paper deals with the geometry of the classes constructed.2000 Mathematics subject classification: primary 30C45.

Abstract. In this paper we consider a class of univalent orientation-preserving harmonic functions defined on the exterior of the unit disk which satisfy the condition ∞ n=1 n p (|an| + |bn|) ≤ 1. We are interested in finding radius of univalence and convexity for such class and we find extremal functions. Convolution, convex combination, and explicit quasiconformal extension for this class are also determined.

In this paper, we shall study estimates for the coefficients an, n = 1,2 of a class of univalent harmonic mappings defined on the exterior of the unit disk Ũ = { z : |z| > 1 }, which keep infinity fixed. For this purpose, we apply Faber polynomials and an inequality of the Grunsky type.