On polyharmonic univalent mappings

In this paper, we introduce a class of complex-valued polyharmonic mappings, denoted by HSp( ), and its subclass HS 0 p( ), where 2 [0; 1] is a constant. These classes are natural generalizations of a class of mappings studied by Goodman in the 1950s. We generalize the main results of Avci and Z lotkiewicz from the 1990s to the classes HSp( ) and HS 0( ), showing that the mappings in HSp( ) are univalent and sense preserving. We also prove that the mappings in HS 0 p( ) are starlike with respect to the origin, and characterize the extremal points of the above classes. AMS 2010 Subject Classication: Primary 30C45; Secondary 30C20, 30C65.