After recalling the definition of hyperbolic distance and Schwarz lemma, the author focuses on ultrahyperbolic and pseudohermitian metric and Ahlfors’ lemmas and on the estimate opposite to Ahlfors–Schwarz lemma, proved by himself. Moreover, he writes about the Hopf differential of a quasiregular harmonic map with respect to strongly negatively curved metric and Ahlfors–Schwarz lemma concerning harmonic-quasiregular maps.