On J. C. C. Nitsche's type inequality for hyperbolic space H³

Let $\mathbf H^3$ be the hyperbolic space identified with the unit ball $\mathbf{B}^3 = \{x\in \mathbf{R}^3: |x| < 1\}$ with the Poincar\'e metric $d_h$ and assume that ${\mathcal{A}}(x_0,p,q):=\{x: p<d_h(x,x_0)< q\}\subset \mathbf H^3$ is an hyperbolic annulus with the inner and outer radii $0<p<q<\infty$. We prove that if there exists a proper hyperbolic harmonic mapping between annuli ${\mathcal{A}}(x_0,a,b)$ and ${\mathcal{A}}(y_0,\alpha,\beta)$ in the hyperbolic space $\mathbf H^3$, then $\beta/\alpha>1+\psi(a,b)$, where $\psi$ is a positive function.