On J. C. C. Nitsche's type inequality for hyperbolic space $\mathbf{H}^3$

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 Poincaré 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 ${\mathcal{A}}(y_0,α,β)$ in the hyperbolic space $\mathbf H^3$, then $β/α>1+ψ(a,b)$, where $ψ$ is a positive function.

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