Sharp mean-width and Jacobian bounds for Euclidean and hyperbolic harmonic maps

We prove a sharp mean-width inequality for monotone zonal operators and derive global and differential bounds for Euclidean and hyperbolic-harmonic self-maps of the unit ball. Let $k:[0,\pi ]\to \mathbb{R}$ be continuous and nonincreasing, and let $T_k$ be the associated zonal integral operator, $$(T_{k}F)(\xi )={\int }_{{\mathbb{S}}^{n-1}}k\text{\hspace{0.17em}⁣}(\arccos ⟨\xi ,\eta ⟩)F(\eta )d\sigma (\eta ),$$ where $\sigma$ is normalized surface measure. For every measurable $F:{\mathbb{S}}^{n-1}\to \overline{{\mathbb{B}}^n}$, we prove $$w\text{\hspace{0.17em}⁣}(\mathrm{co}{T}_{k}F({\mathbb{S}}^{n-1}))\le 2{\lambda }_1(k),$$ where $w$ denotes mean width, normalized by $w(\overline{{\mathbb{B}}^n})=2$, and ${\lambda }_1(k)$ is the eigenvalue of $T_k$ on the space of spherical harmonics of degree one. For strictly decreasing kernels, equality holds exactly for $F(\eta )=Q\eta$ almost everywhere, with $Q\in O(n)$. For the ordinary Poisson kernel, the multiplier is $r$, yielding sharp mean-width, intrinsic-volume, and image-volume contraction. In particular, $\mathrm{\mid }f(r{\mathbb{B}}^n)\mathrm{\mid }\le {\omega }_n{r}^n$ without injectivity assumptions, answering the area and higher-dimensional volume question of Koh and Kovalev.