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.

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In this paper we discuss bounds for the total curvature of nonparametric minimal surfaces by using the properties of planar harmonic mappings.

Let Aut H (D) be the class of harmonic automorphisms of the unit disk D. For f (re iϕ ) = ∞ n=−∞ c n r |n| e inϕ ∈ Aut H (D) the upper bounds of Re n k=1 {c k + c −k } are given.