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In this paper, we investigate maximum principle, convergence of sequences and angular limits for harmonic Bloch mappings. First, we give the maximum principle of harmonic Bloch mappings, which is a generalization of the classical maximum principle for harmonic mappings. Second, by using the maximum principle of harmonic Bloch mappings, we investigate the convergence of sequences for harmonic Bloch mappings. Finally, we discuss the angular limits of harmonic Bloch mappings. We show that the asymptotic values and angular limits are identical for harmonic Bloch mappings, and we further prove a result that applies also if there is no asymptotic value. A sufficient condition for a harmonic Bloch mapping has a finite angular limit is also given. 1. Preliminaries A complex-valued function f is said to be harmonic in a simply connected domain Ω of the complex plane C if and only if both Re{f} and Im{f} are real harmonic mappings in Ω. Every harmonic mapping f in Ω has the canonical decomposition: (1.1) f = h+ g, where both h and g are analytic in Ω and g(z0) = 0 for some prescribed point z0 ∈ Ω (cf. [8]). Let D = {z ∈ C : |z| < 1} and T = {z : |z| = 1}. Throughout this paper, we consider harmonic mappings in D. Definition 1. Let f be a harmonic mapping satisfying the Lipschitz condition when regarded as a function from the hyperbolic disk D into C, endowed with Received July 16, 2013; Revised April 6, 2014. 2010 Mathematics Subject Classification. Primary 30C65, 30C45; Secondary 30C20.

MAXIMUM PRINCIPLE, CONVERGENCE OF SEQUENCES AND ANGULAR LIMITS FOR HARMONIC BLOCH MAPPINGS