Extensions of Radial Monotonicity in the Poincaré Ball: Tangential, Inward, and Variable-Scale Directions

We extend the finite-scale radial monotonicity result for the Poincaré ball to three settings: the inward radial direction, the tangential direction, and the case of a variable Euclidean step \(\varepsilon(r)\). In the inward case we show that the hyperbolic step \(\delta_-(r,\varepsilon)\) is strictly increasing in \(r\) on the domain \(r>2\mathrm{arctanh}(\varepsilon)\). In the tangential case, interpreting the hyperbolic step as the hyperbolic distance to the displaced point, we prove strict monotonicity of \(\delta_\perp(r,\varepsilon)\) on the domain \(a^2+\varepsilon^2<1\), where \(a=\tanh(r/2)\). We then establish a sphere-averaged analogue \(\bar{\delta}(r,\varepsilon)\) on the restricted domain \(\varepsilon<a<1-\varepsilon\) (which requires \(\varepsilon<1/2\)), where pointwise monotonicity holds for every direction of the displacement, and a volume-based characterization \(\delta_{\mathrm{vol}}(r,\varepsilon)\) on the full outward domain \(0\leq r<2\mathrm{arctanh}(1-\varepsilon)\). Finally, for a variable step \(\varepsilon(r)\) we derive a necessary and sufficient criterion for monotonicity of \(\delta_+(r,\varepsilon(r))\), show that it holds automatically whenever \(\varepsilon'(r)\geq0\), and exhibit an explicit counterexample of the form \(\varepsilon(r)=c(1-\|z\|)\) with \(c\in(0,1)\) for which monotonicity fails. The extension of the sphere-averaged result to the full domain \(a+\varepsilon<1\) remains open.