Analytic inheritance of geometric properties of planar harmonic mappings

Planar Harmonic Mappings is a relatively new subfield of Complex Analysis which involves functions that have Laplacian zero but do not necessarily satisfy the Cauchy-Riemann equations, i.e. not necessarily analytic. The field was first explored in-depth by Clunie and Shell-Small in 1984 [3] and has seen steady work since that time.; A basic result states that every planar harmonic mapping can be decomposed into the sum of an analytic and an anti-analytic function. As in the analysis of analytic functions, planar harmonic functions can be classified geometrically by considering the image of the unit disk and divided into subclasses according to certain normalizations.; Once the geometric properties of the subclasses have been defined and illustrated the focus of the rest of the paper will be on the analytic inheritance of a planar harmonic mapping. That is, if the analytic part of a planar harmonic mapping exhibits the same geometric properties as the parent mapping then that mapping would exhibit analytic inheritance. If those properties do not hold, that would provide a counter-example to inheritance. Then we will provide guideline as to when inheritance will hold and when it will not.