WEIGHTED LIPSCHITZ CONTINUITY, SCHWARZ–PICK'S LEMMA AND LANDAU–BLOCH'S THEOREM FOR HYPERBOLIC-HARMONIC MAPPINGS IN ℂN

Shaolin Chen,Saminathan Ponnusamy,Xiantao Wang

doi:10.3846/13926292.2013.756834

Abstract

In this paper, we discuss some properties on hyperbolic-harmonic functions in the unit ball of ℂ n . First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz–Pick type theorem for hyperbolic-harmonic functions and apply it to prove the existence of Landau-Bloch constant for functions in α-Bloch spaces.

Highlights

Introduction and PreliminariesLet Cn denote the complex Euclidean n-space
The h-harmonic Bloch space HB consists of complex-valued hharmonic functions f defined on Bn such that f HB = sup 1 − |z|2 Df (z) + Df (z) < ∞
Our first aim is to characterize the functions in h-harmonic Bloch spaces in terms of their corresponding weighted Lipschitz functions

Summary

Introduction and Preliminaries

Let Cn denote the complex Euclidean n-space. For z = (z1, . . . , zn) ∈ Cn, the conjugate of z, denoted by z, is defined by z = (z1, . . . , zn). The h-harmonic Bloch space HB consists of complex-valued hharmonic functions f defined on Bn such that f HB = sup 1 − |z|2 Df (z) + Df (z) < ∞. Our first aim is to characterize the functions in h-harmonic Bloch spaces in terms of their corresponding weighted Lipschitz functions This is done in Theorem 1 which is a generalization of [11, Theorem 1] and [15, Theorem 3]. For α > 0, the vector-valued h-harmonic α-Bloch space HBn(α) consists of all functions in PH(Bn, Cn) such that f HBn(α) = sup 1 − |z|2 α fz(z) + fz(z). One of the long standing open problems in function theory is to determine the precise value of the univalent Landau-Bloch constant for analytic functions of D In recent years, this problem has attracted much attention, see [4, 18, 20] and references therein. It is worth pointing out that Theorems 2 and 3 are generalizations of [11, Theorem 1] and [9, Theorem 2.4], respectively

Characterization of Mappings in h-Harmonic Bloch Spaces

Schwarz–Pick Type Theorem and Landau–Bloch Theorem