Stable classes of harmonic mappings

Abstract

Let ${\mathcal{H}}_0$ denote the set of all sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\ID$, normalized with $h(0)=g(0)=g^{\prime }(0)=0$ and $h^{\prime }(0)=1$. In this paper, we investigate some properties of certain subclasses of ${\mathcal{H}}_0$, including inclusion relations and stability analysis by precise examples, coefficient bounds, growth, covering and distortion theorems. As applications, we build some Bohr inequalities for these subclasses by means of subordination. Among these subclasses, six classes consist of functions $f=h+\overline{g}\in {\mathcal{H}}_0$ such that $h+\hat{I}\mu g$ is univalent (or convex) in $\D$ for each $|\hat{I}\mu |=1$ (or for some $|\hat{I}\mu |=1$, or for some $|\hat{I}\mu |\le 1$). Simple analysis shows that if the function $f=h+\overline{g}$ belongs to a given class from these six classes, then the functions $h+\overline{\hat{I}\mu g}$ belong to corresponding class for all $|\hat{I}\mu |=1$. We call these classes as stable classes.

15 pages; To appear in Bulletin des sciences mathematiques