Linear connectivity, Schwarz-Pick lemma and univalency criteria for   planar harmonic mappings
  In this paper, we first establish the Schwarz-Pick lemma of higher-order and apply it to obtain a univalency criteria for planar harmonic mappings. Then we discuss distortion theorems, Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images. 
Shaolin Chen, Saminathan Ponnusamy, Antti Rasila and Xiantao Wang
math.CV