Stable geometric properties of pluriharmonic and biholomorphic mappings, and Landau–Bloch’s theorem

Abstract

In this paper, we investigate some properties of pluriharmonic mappings defined in the unit ball. First, we discuss some geometric univalence criteria on pluriharmonic mappings, and then establish a Landau-Bloch theorem for a class of pluriharmonic mappings.

Pluriharmonic Mappings with the Convex Holomorphic Part

“…Every pluriharmonic mapping f: B n ⟶ C n can be written as f � h + g, where h, g are the holomorphic mappings, and this representation is unique if g(0) � 0 (cf. [1][2][3][4][5]).…”

“…By the assumption ‖ω f ‖ ∞ ≤ N < 1, the inverse mapping theorem and Teorem 4, we know that f − 1 is diferentiable (cf. [3]). Moreover, by [3], (28), we have…”

Landau’s Theorem for Solutions of the -Equation in Dirichlet-Type Spaces

“…Unfortunately, there is no analogue of Landau's theorem for general classes of functions (see [4,18]). In order to obtain analogues of Landau's theorem for more general classes of functions, it is necessary to restrict the class of functions considered (compare with [4][5][6][7]18]). The first aim of this paper is to extend the classical Landau theorem to the solutions of (1.1).…”