Let TK(D) be the class of K-quasiconformal automorphisms of a domain D ⊂ ℝn with identity boundary values. Teichmüller’s problem is to determine how far a given point x ∈ D can be mapped under a mapping f∈ TK(D). We estimate this distance between x and f(x) from the above by using two different metrics, the distance ratio metric and the quasihyperbolic metric. We study Teichmüller’s problem for Gromov hyperbolic domains in ℝn with identity values at the boundary of infinity. As applications, we obtain results on Teichmüller’s problem for ψ-uniform domains and inner uniform domains in ℝn.