Stable geometric properties of analytic and harmonic functions (Q2845629)

A planar harmonic mapping in the unit disk \(\mathbb D\) is a complex-valued harmonic function \(f\) which maps \(\mathbb D\) onto some planar domain \(f(\mathbb D)\). The mapping \(f\) has a canonical decomposition \(f=h+\bar g \), where \(h\) and \(g\) are analytic in \(\mathbb D\). A mapping \(f=h+\bar g \) is sense preserving in \(\mathbb D\) if and only if \(h'\) does not vanish in \(\mathbb D\) and the dilatation \(\omega = g'/h'\) has the property \(|\omega|<1\) in \(\mathbb D\).NEWLINENEWLINEThe authors prove that if \(f=h+\bar g\) is a sense preserving harmonic mapping in \(\mathbb D\) then for all \(|\lambda|=1\) the functions \(f_{\lambda}=h+\lambda\bar g\) are univalent (respectively close-to-convex, starlike or convex) if and only if the analytic functions \(F_\lambda=h+\lambda g\) are univalent (respectively close-to-convex, starlike or convex).