Schwarzian derivative, integral means, and the affine and linear invariant families of biharmonic mappings

Abstract

In this paper we discuss the properties of the Schwarzian derivative, integral means and the affine and linear invariant families of biharmonic mappings. First, we introduce the Schwarzian derivative S ( F ) for biharmonic mappings F = ∣ z∣ 2 G + H, and obtain several necessary and sufficient conditions for S ( F ) to be analytic. Second, we introduce the subordination of biharmonic mappings and obtain inequalities for integral means of subordinate biharmonic mappings. Finally, we introduce the affine and linear invariant families of biharmonic mappings and prove several estimates related to the Jacobian of functions in these invariant families.