In this paper, for a general metric density (Formula presented.) , we build a differential inequality to study the hyperbolically partial derivative of (Formula presented.) -harmonic quasiconformal mappings and generalize a result given by Kne z̆ević and Mateljević. As one application, we obtain the hyperbolically (Formula presented.)-biLipschitz continuity of such class of mappings. As another application, we generalize the classical Koebe Theorem to this class of mappings and use this result to study its quasihyperbolically biLipschitz continuity.