Quasiconformal harmonic mappings and close-to-convex domains

Abstract

Let f = h + ? be a univalent sense preserving harmonic mapping of the unit disk U onto a convex domain ?. It is proved that: for every a such that |a| < 1 (resp. |a| = 1) the mapping ?a = h + a? is an |a| quasiconformal (a univalent) close-to-convex harmonic mapping. This gives an answer to a question posed by Chuaqui and Hern?ndez (J. Math. Anal. Appl. (2007)). 2010 Mathematics Subject Classifications. Primary 30C55, Secondary 31C05. .