Geometric Function Theory on Uniformly Quasiconformally Homogeneous Domains

Theorem 1.2. Let X,Y be proper subdomains of $\mathbb{R}^n$ and suppose X is a uniformly M-QCH domain. Then, $f:(X,k_X)\to (Y,k_Y)$ is K-quasiconformal if and only if f is $\eta$ -quasisymmetric, in the respective quasihyperbolic metrics. We then shift our attention to studying the geometric properties of particular classes of mappings. Normal meromorphic functions were introduced by Lehto and Virtanen in [19], and Bloch functions were investigated by Pommerenke in [21]. Recall that a meromorphic function $f: \mathbb{D} \to \overline{\mathbb{C}}$ is normal if $\{f \circ A : A \in G\}$ is a normal family, where G is the set of Möbius maps on $\mathbb{D}$ , and a holomorphic function $g: \mathbb{D} \to \mathbb{C}$ is a Bloch function if the family $\{g(A(z)) - g(A(0)) : A \in G\}$ is a normal family, where G is again the set of Möbius maps on $\mathbb{D}$ . These classes of mappings have been studied at length in the planar setting. Such maps have been generalized to higher dimensions. The generalization for normal meromorphic functions was established by Vuorinen in [25]. The generalization of Bloch functions was systematically explored for the first time by the first author and Nicks in [9]. The first author and Nicks define normal quasiregular mappings which are quasiregular maps $f: X \to S^n$ such that $\{f \circ A: A \in G\}$ is a normal family, where $X \subset S^n$ is a metric space arising from a conformal metric and G is a collection of conformal isometries of X. If the range is instead $\mathbb{R}^n$ then f is normal if the family $\{f(A(x)) - f(A(x_0)): A \in G\}$ is normal for some $x_0 \in X$ . These easily compare to the definitions of normal meromorphic and Bloch functions. However, there are only a few known examples of domains X with a transitive collection of isometries: $X = \mathbb{B}^n$ and G the collection Möbius maps or $X = \mathbb{R}^n$ and G the collection of translations. Consider the following definition.

Theorem 1.5. Let $X \subset S^n$ be a proper subdomain equipped with the quasihyperbolic metric. Furthermore, suppose X is a uniformly M-QCH domain with G as the transitive collection of orientation-preserving M-quasiconformal mappings $A: X \to X$ arising from X. Then, - (i) $f: X \to S^n$ is a normal quasiregular mapping if and only if f is uniformly continuous with respect to $k_X$ and $\sigma$ . - (ii) $f: X \to \mathbb{R}^n$ is a normal quasiregular mapping if and only if f is uniformly continuous with respect to $k_X$ and the Euclidean distance. If in the definition above, we replace $S^n$ or $\mathbb{R}^n$ with a subdomain $Y \subset S^n$ equipped with the quasihyperbolic metric, and Y omits enough points (Rickman's constant), then f is automatically a normal quasiregular mapping by [22, Thm IV.2.1]. Thus, only studying when the range is $S^n$ or $\mathbb{R}^n$ is reasonable. Next, we prove a global Hölder continuity result for normal quasiregular maps when the range is $S^n$ equipped with the spherical metric.

Theorem 1.6. Let $X \subset S^n$ be a proper subdomain equipped with the quasihyperbolic metric. Furthermore, suppose X is a uniformly M-QCH domain with G as the transitive collection of orientation-preserving M-quasiconformal mappings $A: X \to X$ arising from X. Let $f: X \to S^n$ be a K-quasiregular mapping and let $\beta = (KM^2)^{1/(1-n)}$ . Then f is normal if and only if f is globally $\beta$ -Hölder, that is, there exists $C_0 > 0$ such that <span id="page-3-3"></span>(1.1) $$\sigma(f(x), f(y)) \le C_0 k_X(x, y)^{\beta},$$ for all $x, y \in X$ . <span id="page-4-0"></span>A similar result can be shown for the normal quasiregular maps when the range is $\mathbb{R}^n$ .

Theorem 1.7. Let $X \subset S^n$ be a proper subdomain equipped with the quasihyperbolic metric. Suppose X is a uniformly M-QCH domain with G as the transitive collection of orientation-preserving M-quasiconformal mappings $A: X \to X$ arising from X. Let $f: X \to \mathbb{R}^n$ be a K-quasiregular mapping and let $\beta = (KM^2)^{1/(1-n)}$ . Then, f is normal if and only if there exists $C_0 > 0$ such that <span id="page-4-1"></span> $$|f(x) - f(y)| \le C_0 \max\{k_X(x, y), k_X(x, y)^{\beta}\},\$$ for all $x, y \in X$ . While our focus is on uncovering geometric properties of normal quasiregular mappings, it is useful to note some examples and non-examples of such mappings. Recall in [19, Thm 9], Lehto and Virtanen prove a planar meromorphic function cannot be normal in any neighborhood of an isolated essential singularity. For example, there do not exist any normal meromorphic maps in $\mathbb{B}^2 \setminus \{0\}$ equipped with the hyperbolic metric for which 0 is an essential singularity. This result was then generalized to higher dimensions by Heinonen and Rossi in [17, Thm 2.3]. They show if $f: (\mathbb{B}^n \setminus \{0\}, d_{\delta}) \to (S^n, \sigma)$ has an essential singularity at 0, then f cannot be a normal quasiregular map, where the metric $d_{\delta}$ is defined by $$d_{\delta}(x,y) = \inf \int_{\gamma} \delta(s) |ds|,$$ the infimum is taken over all paths $\gamma$ joining x and y in $\mathbb{B}^n \setminus \{0\}$ , and the density $\delta$ is required to be a positive continuous function in $\mathbb{B}^n \setminus \{0\}$ with $\delta(x) = o(|x|^{-1})$ , as $x \to 0$ . The construction of $d_{\delta}$ is meant to mimic the behavior of the hyperbolic metric on $\mathbb{B}^2 \setminus \{0\}$ near the singularity. However, the quasihyperbolic metric is not comparable to $d_{\delta}$ on $\mathbb{B}^n \setminus \{0\}$ , as its density is $O(|x|^{-1})$ as $x \to 0$ . Thus, it would be interesting to see whether a normal quasiregular map from a punctured neighborhood $U \setminus \{x_0\}$ equipped with the quasihyperbolic metric to $S^n$ equipped with the spherical metric requires $x_0$ be a removable singularity or a pole. We leave this as an open question. The paper is organized as follows. In Section 2, we provide necessary preliminary information. Sections 3 and 4 house our main results. Section 3 focuses on investigating the relationship between quasisymmetric and quasiconformal mappings on K-QCH domains, while Section 4 centers on geometric properties of normal quasiregular mappings on K-QCH domains.

Proposition 2.1. ([11], Lemma 1). Let X be a proper subdomain of $\mathbb{R}^n$ . For every $x, y \in X$ there exists a quasi-hyperbolic geodesic $\gamma$ with endpoints x and y.

Proposition 2.2. ([12], Corollary 2.2). If $X \subset \mathbb{R}^n$ proper subdomain of $\mathbb{R}^n$ , then $k_X$ is a complete metric on D, which determines the usual topology there. Before we conclude our discussion on metric spaces, consider the following restriction we can place on a metric space.

Theorem 2.11. ([24], Theorem 30.1). The function $\mathcal{H}_n:(0,\infty)\to\mathbb{R}$ has the following properties: - (a) $\mathcal{H}_n$ is decreasing. - (b) $\lim_{r\to\infty} \mathcal{H}_n(r) = 0$ . - (c) $\lim_{r\to 0} \mathcal{H}_n(r) = \infty$ . - (d) $\mathcal{H}_n(r) > 0$ . We now transition to discussing some properties and results of both quasiregular and quasiconformal maps in the setting of the quasihyperbolic metric. The following result uncovers a local geometric property for quasiregular maps.

Theorem 2.12. ([8], Theorem 1.1). Let E a proper subdomain of $\mathbb{R}^n$ equipped with the Euclidean distance function. Further, suppose $x_0 \in E$ and $f: E \to \mathbb{R}^n$ a non-constant quasiregular map. Then, there exist $C_0 > 1$ and $r_0 > 0$ such that, for all $0 < T_0 \le 1$ and all $r \in (0, r_0)$ , $$T_0^{-\mu} \le \frac{L_f(x_0, r)}{\ell_f(x_0, T_0 r)} \le \frac{C_0^2}{T_0^{-\nu}},$$ where $\mu = (i(x_0, f)/K_I(f)^{1/(n-1)})$ and $\nu = (K_O(f)i(x_0, f))^{1/(n-1)}$ . Moreover, $C_0$ depends only on $n, K_O(f)$ and $i(x_0, f)$ . A corollary to Rickman's well-known $K_O$ -inequality for quasiregular mappings [22, Thm II.2.4] provides a stronger version of the $K_O$ -inequality for quasiconformal mappings.

Proposition 2.13. ([22], Corollary II.2.7). If $f: D \to D'$ is quasiconformal and $\Gamma$ is a path family in D, then $$M(\Gamma) \leq K_O(f)M(f(\Gamma)),$$ where $f(\Gamma)$ denotes the image of $\Gamma$ under f. Focusing on quasiconformal maps allows us more insight into the behavior of the map locally and globally. In the following result, Gehring and Osgood [12] reveal that quasiconformal maps on domains equipped with the quasihyperbolic metric are globally Lipschitz but locally Hölder.

Theorem 2.14. Let $X, Y \subset \mathbb{R}^n$ be domains and $f: X \to Y$ an M-quasiconformal map. Then, there exist constants $C_1, C_2 > 0$ depending only on n and M such that $$k_Y(f(x_1), f(x_2)) \le C_1 \max\{k_X(x_1, x_2), k_X(x_1, x_2)^{\frac{1}{\alpha}}\}$$ and $$k_Y(f(x_1), f(x_2)) \ge C_2 \min\{k_X(x_1, x_2), k_X(x_1, x_2)^{\alpha}\}$$ for all $x_1, x_2 \in X$ , where $\alpha = M^{\frac{1}{n-1}}$ . 2.3. Quasisymmetry and Weak Quasisymmetry. Recall that a homeomorphism between metric spaces $f:(X,d_X)\to (Y,d_Y)$ is L-bi-Lipschitz if there exists $L\geq 1$ such that for all $x,y\in X$ we have $\frac{1}{L}d_X(x,y)\leq d_Y(f(x),f(y))\leq Ld_X(x,y)$ . Bi-Lipschitz maps can only distort distances by a bounded amount. To consider a more general class of mappings, we want to allow bounded distortion of relative distances instead.

Proposition 2.17. Let $f:(X,d_X) \to (Y,d_Y)$ be a map. Then f is weakly H-quasisymmetric if and only if for every $x \in X$ and r > 0 we have $f(B_X(x,r)) \subset B_Y(f(x),Hl_f(x,r))$ . Proof. To begin, let us assume f is weakly H-quasisymmetric. Let x ∈ X, r > 0, and v ∈ f(BX(x, r)). Then, there exists y ∈ X such that f(y) = v. Choose z ∈ X such that d(x, z) = r and d^Y (f(x), f(z)) = ℓ^f (x, r). If d^X (x, y) ≤ r = d^X (x, z), weak H-quasisymmetry of f implies $$d_Y(f(x), f(y)) \le Hd_Y(f(x), f(z)) = H\ell_f(x, r).$$ Hence, v ∈ B^Y (f(x), Hℓ^f (x, r)). Conversely, let x, y, z ∈ X be distinct points such that dX(x, y) ≤ d^X (x, z). Set r = d^X (x, z). Then, x, y ∈ B^X (x, r). By assumption, f(x), f(y) ∈ f(BX(x, r)) ⊂ B^Y (f(x), Hℓ^f (x, r)), so $$d_Y(f(x), f(y)) \le H\ell_f(x, r).$$ As d^Y (f(x), f(z)) ≥ ℓ^f (x, r), we have $$d_Y(f(x), f(y)) \le H\ell_f(x, r) \le Hd_Y(f(x), f(z)),$$ so f is weakly H-quasisymmetric. 2.4. Families of Continuous Functions. Let X, Y be subsets of S ⁿ and d^X , d^Y conformal distance functions. We define C(X, Y ) as the set of all continuous functions from X to Y. Now, we can make C(X, Y ) into a metric space with the distance function defined the following way: let (Ki)^∞ ⁱ=0 be a compact exhaustion of X and for f, g ∈ C(X, Y ) set, $$d_i(f,g) = \sup\{d_Y(f(x),g(x)) : x \in K_i\}$$ and $$d_{X,Y}(f,g) = \sum_{i=1}^{\infty} \frac{1}{2^i} \left( \frac{d_i(f,g)}{1 + d_i(f,g)} \right)$$ . Then, d^X (fm, f) → 0 if and only if f^m → f uniformly on compact subsets of X. The topology which arises from this distance function is called the topology of uniform convergence on compact sets. It will be of importance to discuss the relative compactness of families of functions. We say that a family of functions F ⊂ C(X, Y ) is relatively compact in C(X, Y ) if its closure is compact. Just above we saw that C(X, Y ) can be viewed as a metric space, and sequential compactness coincides with compactness in metric spaces. Therefore, F is relatively compact if and only any sequence f^m ∈ F has a subsequence which converges uniformly on compact subsets to an element of C(X, Y ). This concept will be key when combined with ideas from the following definitions and results. Definition 2.18. Let ω : [0,∞) → [0,∞) be a continuous increasing function with ω(0) = 0. We call a map f : X → Y ω-continuous if f has modulus of continuity ω, that is, if $$d_Y(f(x), f(y)) \le \omega(d_X(x, y))$$ for all x, y ∈ X. Definition 2.19. A family F ⊂ C(X, Y ) is called: (i) uniformly ω-continuous if there exists L > 0 such that for all x, y ∈ X and all f ∈ F, $$d_Y(f(x), f(y)) \le L\omega(d_X(x, y));$$ (ii) uniformly ω-continuous on compact sets if for each compact set E ⊂ X, there exists L > 0 such that for all x, y ∈ E and all f ∈ F, $$d_Y(f(x), f(y)) \le L(d_X(x, y));$$ (iii) locally uniformly ω-continuous if for each x⁰ ∈ X, there exists r, L > 0 such that for all x, y ∈ BX(x0, r) and all f ∈ F, $$d_Y(f(x), f(y)) \le L\omega(d_X(x, y)).$$ <span id="page-10-0"></span>The following proposition tells us that parts (ii) and (iii) of the above definition are equivalent. Proposition 2.20 ([\[9\]](#page-26-6), Proposition 2.6). Let X, Y subdomains of S n equipped distance functions d^X , d^Y , respectively both arising from conformal metrics. A family F ⊂ C(X, Y ) is locally uniformly ω-continuous if and only if it is uniformly ω-continuous on compact sets. Recall the following definition. Definition 2.21. Let X, Y subdomains of S ⁿ with respective distance functions d^X , d^Y each arising from a conformal metric. Let F ⊂ C(X, Y ). - (i) F is equicontinuous at x⁰ ∈ X if for every ε > 0 there exists δ > 0 such that d^Y (f(x), f(x0)) < ε whenever dX(x, x0) < δ and f ∈ F. - (ii) F is equicontinuous on X if it is equicontinuous at each point x⁰ ∈ X. <span id="page-10-1"></span>We now consider a version of the Arzela-Ascoli theorem for our setting([\[20,](#page-27-15) Theorem 47.1]). Theorem 2.22. Let F be a family of continuous functions from a locally compact Hausdorff metric space X to a metric space Y. Then F is relatively compact in C(X, Y ) if and only if - (i) the family F is equicontinuous on X and - (ii) for every x ∈ X, the orbit F(x) = {f(x) : f ∈ F} is relatively compact in Y. Viewing C(X, Y ) as a metric space allows us to reinterpret the Arzela-Ascoli theorem for families F ⊂ C(X, Y ) which are locally uniformly ω-continuous. This modified version of Arzela-Ascoli provides us with geometric insight that will be of use in Section 4. - <span id="page-10-2"></span>Theorem 2.23 ([\[9\]](#page-26-6),Theorem 2.9). Let X, Y be subdomains of S ⁿ with conformal metrics and associated distance functions d^X , d^Y respectively and suppose the metric on Y is complete. Let F ⊂ C(X, Y ) be locally uniformly ω-continuous. Then F is relatively compact in C(X, Y ) if and only if there exists x⁰ ∈ X such that F(x0) = {f(x0) : f ∈ F} is relatively compact in Y. - 2.5. Normal Families. As stated in our introduction, Beardon and Minda [\[3\]](#page-26-8) characterize normal families in terms of a locally uniform Lipschitz condition with a view to applications to families of holomorphic functions. The first author and Nicks [\[9\]](#page-26-6) then generalized the Beardon and Minda viewpoint to the higher dimensional setting, and a local uniform H¨older condition with a view to applications to families of quasiregular maps. Here we take a moment to highlight some of this work, as we will be relying on it in Section 4. Definition 2.24. If X, Y are subdomains of S n , then for M ≥ 1, denote by QM(X, Y ) the subset of C(X, Y ) consisting of all M-quasiregular mappings from X to Y. Now, recall the following definition of a family being normal relative to a domain from [\[3\]](#page-26-8).

Theorem 2.27. ([9], Theorem 3.7). Let $X \subset S^n$ be a domain equipped with distance function $d_X$ arising from a conformal metric and let $\mathcal{F} \subset \mathcal{Q}_M(X,Y)$ be a family of M-quasiregular mappings defined on X with image contained in $Y \subset S^n$ equipped with distance function $d_Y$ arising from a complete conformal metric. Then $\mathcal{F}$ is relatively compact in C(X,Y) if and only if - (i) $\mathcal{F}$ is locally uniformly $\omega$ -continuous, with $\omega(t) = t^{\alpha}$ , $\alpha = M^{1/(1-n)}$ and - (ii) there exists $x_0 \in X$ such that $\mathcal{F}(x_0) = \{f(x_0) : f \in \mathcal{F}\}$ is relatively compact in Y. In particular, if Y is $S^n$ with the spherical metric and if $\mathcal{F} \subset \mathcal{Q}_M(X,S^n)$ is a family of M-quasiregular mappings, then $\mathcal{F}$ is normal if and only if $\mathcal{F}$ is locally uniformly $\omega$ -continuous, with $\omega(t) = t^{\alpha}, \alpha = M^{1/(1-n)}$ . If we consider a family $\mathcal{F} \subset \mathcal{Q}_M(X,\mathbb{R}^n)$ instead then we can say $\mathcal{F}$ is normal if and only if every sequence in $\mathcal{F}$ has a subsequence which converges uniformly on compact sets to either an element of $\mathcal{Q}_M(X,\mathbb{R}^n)$ or diverges to infinity. From this realization, we form a new definition.

Lemma 3.1. Let $X,Y \subset \mathbb{R}^n$ domains. Suppose $f:(X,k_X) \to (Y,k_Y)$ is a K-quasiconformal map, T>0, and $x_0 \in X$ . Then there exists a constant $\xi$ depending on $x_0,T,n$ , and K such that $$\frac{L_f(x_0, r)}{\ell_f(x_0, Tr)} \le \xi(x_0, n, K, T),$$ for all r > 0. Proof. Let $x_0 \in X$ . Given r, T > 0 find $y, z \in X$ such that $k_X(x_0, y) = r$ , $L_f(x_0, r) = k_Y(f(x_0), f(y))$ , $k_X(x_0, z) = Tr$ and $\ell_f(x_0, Tr) = k_Y(f(x_0), f(z))$ . As the Euclidean and quasihyperbolic distance functions both arise from conformal metrics, they are locally equivalent. So, we can use Theorem 2.12 in the quasihyperbolic setting to say there exist $C_0$ and $r_0 > 0$ such that <span id="page-12-2"></span>(3.1) $$\frac{k_Y(f(x_0), f(y))}{k_Y(f(x_0), f(z))} = \frac{L_f(x_0, r)}{\ell_f(x_0, Tr)} \le \frac{C_0^2}{T^{\nu}},$$ for all $0 < T \le 1$ and $r \in (0, r_0)$ , where $\nu = (K_0(f)i(x_0, f))^{1/(n-1)}$ . In other words, we have the upper bound above if $y, z \in \overline{B_X(x_0, r_0)}$ . We now need to consider when y or z are in $X \setminus \overline{B_X(x_0, r_0)}$ . To do so, we need two cases depending on whether $r_0 < 1$ or $r_0 \ge 1$ . These two cases each have two sub-cases considering whether T < 1 or $T \ge 1$ . Case 1: Suppose $r_0 < 1$ and T < 1. Then Tr < r. We require y or z to be in $X \setminus \overline{B_X(x_0, r_0)}$ , so we can deduce that $k_X(x_0, y) = r > r_0$ . Recall that $\alpha = K^{1/(n-1)}$ . Since $r_0 < r$ and $\frac{1}{\alpha} - 1 < 0$ , notice <span id="page-12-0"></span>(3.2) $$r^{\frac{1}{\alpha}} = r^{\frac{1}{\alpha} - 1} \cdot r \le r_0^{\frac{1}{\alpha} - 1} \cdot r.$$ Moreover, $r_0^{\frac{1}{\alpha}-1} > 1$ which implies $r_0^{\frac{1}{\alpha}-1} \cdot r \ge r$ . Observe, <span id="page-12-1"></span> $$(3.3) (Tr)^{\alpha} = (Tr)^{\alpha-1} \cdot Tr \ge (Tr_0)^{\alpha-1} \cdot Tr.$$ Since $T, r_0 < 1$ , it follows that $$(Tr_0)^{\alpha - 1} \cdot Tr < Tr.$$ Using Theorem 2.14, (3.2), and (3.3) $$\frac{k_{Y}(f(x_{0}), f(y))}{k_{Y}(f(x_{0}), f(z))} \leq \frac{C_{1} \max\{k_{X}(x_{0}, y), k_{X}(x_{0}, y)^{\frac{1}{\alpha}}\}}{C_{2} \min\{k_{X}(x_{0}, y), k_{X}(x_{0}, y)^{\alpha}\}}$$ $$= \frac{C_{1} \max\{r, r^{\frac{1}{\alpha}}\}}{C_{2} \min\{Tr, (Tr)^{\alpha}\}}$$ $$\leq \frac{C_{1} \max\{r, r^{\frac{1}{\alpha}-1}_{0}r\}}{C_{2} \min\{Tr, (Tr_{0})^{\alpha-1}Tr\}}$$ $$= \frac{C_{1}r^{\frac{1}{\alpha}-1}_{0}r}{C_{2}(Tr_{0})^{\alpha-1}Tr}$$ $$= \frac{C_{1}r^{\frac{1}{\alpha}-\alpha}_{0}}{C_{2}T^{\alpha}}.$$ (3.4) <span id="page-13-3"></span>Case 2: Consider when $r_0 < 1$ and $T \ge 1$ . Then Tr > r. We require y or z to not be in $\overline{B_X(x_0, r_0)}$ , so $k_X(x_0, z) = Tr > r_0$ . Notice, <span id="page-13-0"></span> $$(3.5) (Tr)^{\alpha} = (Tr)^{\alpha-1} \cdot Tr \ge r_0^{\alpha-1} \cdot Tr.$$ Since $r > \frac{r_0}{T}$ and $\frac{1}{\alpha} - 1 < 0$ , observe that <span id="page-13-1"></span>(3.6) $$r^{\frac{1}{\alpha}} = r^{\frac{1}{\alpha} - 1} r \le \left(\frac{r_0}{T}\right)^{\frac{1}{\alpha} - 1} \cdot r.$$ Finally, note that $r_0^{\alpha-1} < 1$ and $\left(\frac{r_0}{T}\right)^{\frac{1}{\alpha}-1} > 1$ since $\frac{r_0}{T} < r_0 < 1$ . Then, by Theorem 2.14, (3.5) and (3.6), we have $$\frac{k_{Y}(f(x_{0}), f(y))}{k_{Y}(f(x_{0}), f(z))} \leq \frac{C_{1} \max\{k_{X}(x_{0}, y), k_{X}(x_{0}, y)^{\frac{1}{\alpha}}\}}{C_{2} \min\{k_{X}(x_{0}, y), k_{X}(x_{0}, y)^{\alpha}\}}$$ $$= \frac{C_{1} \max\{r, r^{\frac{1}{\alpha}}\}}{C_{2} \min\{Tr, (Tr)^{\alpha}\}}$$ $$\leq \frac{C_{1} \max\{r, \left(\frac{r_{0}}{T}\right)^{\frac{1}{\alpha}-1}r\}}{C_{2} \min\{Tr, r^{\alpha-1}_{0}Tr\}}$$ $$\leq \frac{C_{1} \left(\frac{r_{0}}{T}\right)^{\frac{1}{\alpha}-1}r}{C_{2}r^{\alpha-1}_{0}Tr}$$ $$= \frac{C_{1}r^{\frac{1}{\alpha}-\alpha}}{C_{2}T^{\frac{1}{\alpha}}}.$$ (3.7) <span id="page-13-4"></span>Case 3: Suppose that $r_0 \ge 1$ and T < 1. Then, Tr < r so we can deduce that $k_X(x_0, y) = r > r_0 \ge 1$ . Then, <span id="page-13-2"></span>(3.8) $$r^{\frac{1}{\alpha}} = r^{\frac{1}{\alpha} - 1} \cdot r \le r_0^{\frac{1}{\alpha} - 1} \cdot r.$$ Assuming $r_0 \ge 1$ implies $r_0^{\frac{1}{\alpha}-1} < 1$ so $r_0^{\frac{1}{\alpha}-1} \cdot r < r$ . Also, since $r > r_0$ , <span id="page-14-0"></span> $$(3.9) (Tr)^{\alpha} = (Tr)^{\alpha-1} \cdot Tr \ge (Tr_0)^{\alpha-1} \cdot Tr.$$ Using Theorem 2.14, (3.8), and (3.9) we have $$\frac{k_Y(f(x_0), f(y))}{k_Y(f(x_0), f(z))} \le \frac{C_1 \max\{k_X(x_0, y), k_X(x_0, y)^{\frac{1}{\alpha}}\}}{C_2 \min\{k_X(x_0, y), k_X(x_0, y)^{\alpha}\}}$$ $$= \frac{C_1 \max\{r, r^{\frac{1}{\alpha}}\}}{C_2 \min\{Tr, (Tr)^{\alpha}\}}$$ $$\le \frac{C_1 \max\{r, (r_0)^{\frac{1}{\alpha} - 1}r\}}{C_2 \min\{Tr, (Tr_0)^{\alpha - 1}Tr\}}$$ $$= \frac{C_1 r}{C_2 \min\{Tr, (Tr_0)^{\alpha - 1}Tr\}}.$$ If $Tr_0 < 1$ then <span id="page-14-3"></span>(3.10) $$\frac{C_1 r}{C_2 \min\{Tr, (Tr_0)^{\alpha - 1} Tr\}} = \frac{C_1 r}{C_2 (Tr_0)^{\alpha - 1} Tr} = \frac{C_1}{C_2 T^{\alpha} r_0^{\alpha - 1}}.$$ Alternatively, if $Tr_0 \geq 1$ then <span id="page-14-4"></span>(3.11) $$\frac{C_1 r}{C_2 \min\{Tr, (Tr_0)^{\alpha - 1} Tr\}} = \frac{C_1 r}{C_2 Tr} = \frac{C_1}{C_2 T}.$$ Case 4: Suppose $r_0 \ge 1$ and $T \ge 1$ . Then, Tr > r so $k_X(x_0, z) = Tr > r_0$ . Notice, <span id="page-14-1"></span> $$(3.12) (Tr)^{\alpha} = (Tr)^{\alpha - 1} Tr \ge (r_0)^{\alpha - 1} Tr.$$ We assumed $r > \frac{r_0}{T}$ , so <span id="page-14-2"></span>(3.13) $$r^{\frac{1}{\alpha}} = r^{\frac{1}{\alpha} - 1} \cdot r \le \left(\frac{r_0}{T}\right)^{\frac{1}{\alpha} - 1} \cdot r.$$ Thus, by Theorem 2.14, (3.12), and (3.13) $$\frac{k_Y(f(x_0), f(y))}{k_Y(f(x_0), f(z))} \le \frac{C_1 \max\{k_X(x_0, y), k_X(x_0, y)^{\frac{1}{\alpha}}\}}{C_2 \min\{k_X(x_0, y), k_X(x_0, y)^{\alpha}\}}$$ $$= \frac{C_1 \max\{r, r^{\frac{1}{\alpha}}\}}{C_2 \min\{Tr, (Tr)^{\alpha}\}}$$ $$\le \frac{C_1 \max\{r, \left(\frac{r_0}{T}\right)^{\frac{1}{\alpha} - 1} r\}}{C_2 \min\{Tr, (r_0)^{\alpha - 1} Tr\}}$$ $$= \frac{C_1 \max\{r, \left(\frac{r_0}{T}\right)^{\frac{1}{\alpha} - 1} r\}}{C_2 Tr},$$ where the last line follows because $r_0^{\alpha-1}Tr \geq Tr$ since $Tr > r_0 \geq 1$ in this case. Now, if $r_0 < T$ , then $\left(\frac{r_0}{T}\right)^{\frac{1}{\alpha}-1} \cdot r > r$ , so <span id="page-15-0"></span>(3.14) $$\frac{C_1 \max\{r, \left(\frac{r_0}{T}\right)^{\frac{1}{\alpha}-1}r\}}{C_2 T r} = \frac{C_1 \left(\frac{r_0}{T}\right)^{\frac{1}{\alpha}-1}r}{C_2 T r} = \frac{C_1 r_0^{\frac{1}{\alpha}-1}}{C_2 T^{\frac{1}{\alpha}}}.$$ If instead $r_0 \geq T$ , then $\left(\frac{r_0}{T}\right)^{\frac{1}{\alpha}-1} \cdot r \leq r$ . Thus, <span id="page-15-1"></span>(3.15) $$\frac{C_1 \max\{r, \left(\frac{r_0}{T}\right)^{\frac{1}{\alpha}} r\}}{C_2 T r} = \frac{C_1 r}{C_2 T r} = \frac{C_1}{T C_2}.$$ Hence, for all $y, z \in X$ such that $k_X(x_0, y) = r$ and $k_X(x_0, z) = Tr$ , by (3.1), (3.4), (3.7), (3.10), (3.11), (3.14) and (3.15), $$\frac{L_f(x_0, r)}{\ell_f(x_0, Tr)} \le \max \left\{ \frac{C_0^2}{T^{\nu}}, \frac{C_1 r_0^{\frac{1}{\alpha} - \alpha}}{C_2 T^{\alpha}}, \frac{C_1 r_0^{\frac{1}{\alpha} - \alpha}}{C_2 T^{\frac{1}{\alpha}}}, \frac{C_1}{C_2 T}, \frac{C_1 r_0^{\frac{1}{\alpha} - 1}}{C_2 T^{\frac{1}{\alpha}}}, \frac{C_1}{C_2 T^{\alpha}}, \frac{C_1}{C_2 T^{\alpha}} \right\} = \xi(x_0, n, K, T),$$ and clearly $\xi$ only depends on $K, n, x_0$ , and T. <span id="page-15-2"></span>Next, we use the preceding lemma to prove quasiconformal maps are weak quasisymmetric.

Theorem 3.2. Let X, Y proper subdomains of $\mathbb{R}^n$ and suppose X is a uniformly M-QCH domain. If $f:(X,k_X)\to (Y,k_Y)$ is K-quasiconformal then it is weakly H-quasisymmetric.

Theorem 3.3. Suppose $(X, d_X)$ and $(Y, d_Y)$ are geodesic metric spaces. Then a homeomorphism $f: X \to Y$ is weakly H-quasisymmetric if and only if f is quasisymmetric.

Definition 1.1. Let $X \subset \mathbb{R}^n$ be a domain and let $K \geq 1$ . We call X a uniformly K-quasiconformally homogeneous domain, or uniformly K-QCH domain, if there exists a collection G of K-quasiconformal mappings which act transitively on X. In other words, for any $x, y \in X$ there exists $g \in G$ such that g(x) = y. Note that G above is not guaranteed to be a group, just a collection. Liouville's Theorem tells us the class of Möbius maps are the only conformal maps in $\mathbb{R}^n$ for $n \geq 3$ , so requiring quasiconformal maps as our transitive collection is the appropriate generalization. It was shown in [12, Lemma 3.2] that every proper domain admits a collection of quasiconformal maps which act transitively, but such a collection is not guaranteed to have a uniform bound on the maximal dilatations. Requiring all maps in the collection to be K-quasiconformal for $K \geq 1$ , as in the definition above, upgrades our domain, allowing for more control over geometric properties. Obviously $\mathbb{B}^n$ is a uniformly 1-QCH domain, as $\mathbb{B}^n$ carries the transitive group of Möbius self maps, but let us discuss some non-trivial examples of uniformly QCH domains. Gehring and Palka proved that there exists a uniformly K-QCH domain whose complement is a Cantor set [12, Example 4.4]. They also have shown that for each $n \geq 2$ there is a domain $D \subset \mathbb{R}^n$ which is uniformly quasiconformally homogeneous such that the complement of D has at least two non degenerate components [12, Example 4.6]. The authors of [5] investigate the properties hyperbolic manifolds require in order to be quasiconformally homogeneous domains, proving that if $n \geq 3$ , a hyperbolic n-manifold is uniformly quasiconformally homogeneous if and only if it is a regular cover of a closed hyperbolic orbifold. This work is continued in [4], where it is proven that any closed hyperbolic surface admitting a conformal automorphism with "many" fixed points is uniformly quasiconformally homogeneous, with constant uniformly bounded away from 1. Other than acquiring examples of uniformly QCH domains, two other lucrative research areas in this field are determining bounds for K and proving various results for domains that are assumed to be uniformly K-QCH ([6], [12], and [13]). In addition to defining uniformly QCH domains, Gehring and Palka defined the quasihyperbolic metric in [12], which generalizes properties of the hyperbolic metric but is for a proper subdomain X of $\mathbb{R}^n$ and is denoted $k_X$ . This is a conformal metric with density $\frac{1}{d(X,\partial X)}$ . Many "hyperbolic-like" properties are preserved by the quasihyperbolic metric [14], but we are not guaranteed X has a large collection of conformal, transitive, quasihyperbolic isometries. Requiring X to also be a uniformly K-QCH domain gives us a domain with a large collection of transitive, K-quasiconformal maps. Moreover, the K-quasiconformal maps which arise from the domain have a global geometric property on X [11]. Thus, uniformly QCH domains equipped with the quasihyperbolic metric correctly generalize $\mathbb{B}^n$ equipped with the hyperbolic metric and the transitive collection of conformal, hyperbolic isometries that arise from it. Gehring and Palka's original intention for defining uniformly QCH domains and equipping them with their quasihyperbolic metric was in hopes of gaining a new characterization for domains which are quasiconformally equivalent to the unit ball. In other words, they desired to find domains for which we could have a truly generalized Riemann Mapping Theorem. This vein of research is still pursued by some ([4],[6], and [7]) but this will not be our area of focus. In this paper, we aim to study the geometric properties of quasiregular mappings on uniformly QCH domains equipped with the quasihyperbolic metric. 1.2. Statement of Results. We investigate the relationship between quasisymmetric and quasiconformal mappings. While quasiconformality is a local property that concerns the distortion of small circles, quasisymmetry is a global three point condition for maps between metric spaces which preserves the relative size of sets. See Section 2 for full definitions. Since the 1960's, we have known that quasiconformal and quasisymmetric mappings are equivalent when the domain and codomain are $\mathbb{R}^n$ for $n \geq 2$ [10]. The same equivalence does not hold for other subdomains in general. For example, in [18, p. 135], it was shown that a conformal map from the unit disk to the slit disk in the complex plane is 1-quasiconformal, but is not quasisymmetric in the Euclidean metric. However, Tyson [23] proved a quasisymmetric map is quasiconformal if the metric spaces of interest are locally compact and connected Ahlfors Q-regular spaces with Hausdorff dimension greater than 1. As for the converse question, much has been done to prove quasiconformal maps are quasisymmetric in metric spaces with properties similar to that of Euclidean space. In [15], Heinonen proved a quasiconformal map from a bounded uniform domain onto a bounded, linearly locally connected domain in $\mathbb{R}^n$ is quasisymmetric. Heinonen also states that analogous results can be proven using the interior metric for the domain. Moreover, Heinonen and Kosela [16] proved that if X and Y are Ahlfors Q-regular metric spaces with Q > 1, X a Loewner space, Y a linearly locally connected space, and $f: X \to Y$ quasiconformal, then $f: X \to Y$ is quasisymmetric. These results in the Euclidean setting are useful. However, Ackermann and the first author noticed that although in Q-regular metric spaces balls of radius r are comparable to $r^Q$ , this is not true in hyperbolic space [1]. The size of the balls will grow exponentially, so the argument made in [16] does not hold in the hyperbolic setting. In [1], they fill this gap by proving that a quasiconformal map $f: \mathbb{B}^n \to \mathbb{B}^n$ is in fact quasisymmetric, where $\mathbb{B}^n$ is equipped with the hyperbolic metric. Moreover, the notions of quasisymmetry and quasiconformality coincide in the hyperbolic setting. We first aim to generalize Ackermann and the first author's work. By requiring X to be a uniformly K-QCH domain equipped with the quasihyperbolic metric we gain a transitive collection of K-quasiconformal mappings with a desirable geometric property. This is a reasonable choice, as such domains properly generalize the transitive collection of conformal hyperbolic isometries that we are afforded in the hyperbolic setting.

Definition 1.3. If X, Y subdomains of $S^n$ , then denote by $\mathcal{Q}_K(X,Y)$ the subset of C(X,Y) consisting of all K-quasiregular mappings from X to Y for $K \geq 1$ . Then, we can generalize the definition of a normal quasiregular map, by loosening the restriction that G be a transitive collection of isometries. We accomplish this by letting our domain be a uniformly quasiconformally homogeneous domain.

Definition 1.4. Let $n \geq 2$ . Let $(X, d_X)$ be a metric space arising from a conformal metric, and suppose X is a uniformly M-QCH domain. Further, let G be the transitive collection of orientation-preserving M-quasiconformal mappings $A: X \to X$ arising from X. (i) We say that a K-quasiregular map $f: X \to S^n$ is a normal quasiregular map into $S^n$ if the family $$\mathcal{F} = \{ f \circ A : A \in G \} \subset \mathcal{Q}_{KM}(X, S^n)$$ is a normal family. (ii) We say that a K-quasiregular map $f: X \to \mathbb{R}^n$ is a normal quasiregular map into $\mathbb{R}^n$ if the family $$\mathcal{F} = \{ f(A(x)) - f(A(x_0)) : A \in G \} \subset \mathcal{Q}_{KM}(X, \mathbb{R}^n)$$ is normal for some $x_0 \in X$ . For the remainder of the paper, when we refer to normal quasiregular mappings, we are discussing this generalized version in Definition 1.4. We are then able to prove geometric results about such mappings analogous to those acquired in [9]. The first of these refers to uniform continuity.

Definition 2.3. A path in $\mathbb{R}^n$ is a continuous mapping $\gamma:[a,b]\to\mathbb{R}^n$ . If $a=t_0\leq t_1\leq\ldots\leq t_m=b$ is a partition of [a,b]. The supremum of the sums $\sum_{i=1}^m |\gamma(t_i)-\gamma(t_{i-1})|$ over all partitions is called the length of $\gamma$ , denoted $\ell(\gamma)$ . If $\ell(\gamma)<\infty$ then $\gamma$ is called a rectifiable path.

Definition 2.4. Let $(X, d_X)$ be a metric space. Then X is c-quasiconvex, for $c \ge 1$ , if each pair of points $x, y \in X$ can be joined by a rectifiable path $\gamma$ such that the length of the path $\ell(\gamma) \le cd_X(x, y)$ . <span id="page-5-1"></span>Remark 2.5. If $(X, d_X)$ is a geodesic metric space then for all x, y we can find a geodesic $\gamma$ with endpoints x and y, and $\ell(\gamma) = d_X(x, y)$ . So, $(X, d_X)$ is also 1-quasiconvex. In particular, Proposition 2.1 implies $(X, k_X)$ is 1-quasiconvex. 2.2. Quasiregular Mappings. Quasiregular mappings in $\mathbb{R}^n$ are a natural generalization of holomorphic functions in the plane. We provide an outline of necessary definitions for quasiregular and quasiconformal mappings below, but refer to Rickman's monograph [22] and Väisälä's lectures [24] for more details. We begin with the definition of a quasiregular map and an equivalent characterization of a quasiconformal map. Then, we provide a few tools and results which will be of use later on.

Definition 2.6. (Analytic Definition) Let $U \subset \mathbb{R}^n$ be a domain. A map $f: U \to \mathbb{R}^n$ is said to be K-quasiregular if f is $ACL^n$ and if there exists a constant $K \geq 1$ such that $$|f'(x)|^n \le KJ_f(x),$$ almost everywhere in U. The smallest K for which this holds is called the outer dilatation of f, denoted $K_O(f)$ . Moreover, if f is K-quasiregular, then $$J_f(x) \le K' \min_{|h|=1} |f'(x)h|^n,$$ where K' is the smallest value for which the inequality holds and is called the inner dilatation, denoted $K_I(f)$ . The maximal dilatation is $K(f) = \max\{K_O(f), K_I(f)\}$ . If f is quasiregular and also injective, then f is called quasiconformal. As quasiregular mappings are not required to be injective, it is useful to determine whether they are injective on "small scales". The local index of a quasiregular map f at $x_0 \in \mathbb{R}^n$ is $$i(x_0, f) = \inf_{U} \sup_{x} \operatorname{card}(f^{-1}(x) \cap U),$$ where the infimum is taken over all neighborhoods U of $x_0$ . If $i(x_0, f) = 1$ then f is called locally injective at $x_0$ . A point $x_0$ where $i(x_0, f) > 1$ is called a branch point of f. Since we will be working with metric spaces, considering the metric definition of quasiconformal mappings will be of use to us. Let us define some preliminary notation. Let $f:(X,d_X)\to (Y,d_Y)$ be a homeomorphism, $x_0\in X$ and r>0 be such that $\overline{B(x_0,r)}\subset X$ . Then, we define the linear dilatations $$L_f(x_0, r) = \sup\{d_Y(f(x), f(x_0)) : d_X(x, x_0) = r\},\$$ $$\ell_f(x_0, r) = \inf\{d_Y(f(x), f(x_0)) : d_X(x, x_0) = r\}.$$ We see that $L_f(x_0, r)$ is measuring the maximal stretch of $\partial B_X(x_0, r)$ by f and $\ell_f(x_0, r)$ measures the minimal stretch. This idea is key to proving one of our main results later on. However, for now, we can use this notion of maximal and minimal stretch to formulate an equivalent definition of a quasiconformal map.

Definition 2.7. (Metric Definition) With the notation as above, the linear dilatation of f at $x_0$ is $$H_f(x_0) = \limsup_{r \to 0} \frac{L_f(x_0, r)}{\ell_f(x_0, r)}.$$ Then, f is K-quasiconformal if and only if $||H_f||_{\infty} \leq K$ for some $K \geq 1$ . The analytic and metric definitions of quasiconformal maps were proven to be equivalent in [10, Corollary 4]. However, there is yet another characterization of quasiconformal mappings which uses modulus of path families. Although this interpretation of quasiconformal maps will not be of use to us, modulus of path families serve as a crucial tool in proving one of our main results. Thus, we take a moment to define the modulus of path families and note some important properties A path family, denoted $\Gamma$ , is a collection of paths $\gamma$ in $\mathbb{R}^n$ . Let $F(\Gamma)$ be the set of all non-negative Borel functions $\rho: \mathbb{R}^n \to \mathbb{R}$ such that $$\int_{\gamma} \rho \ dm \ge 1$$ for every rectifiable curve $\gamma \in \Gamma$ , where m is the Lebesgue measure. Now, we can properly define the modulus of a path family below.

Definition 2.8. The modulus of $\Gamma$ , denoted $M(\Gamma)$ is defined by $$\inf_{\rho \in F(\Gamma)} \int_{\mathbb{R}^n} \rho^n dm,$$ where m is Lebesgue measure. If $F(\Gamma) = \emptyset$ , then we set $M(\Gamma) = \infty$ . Next, we acknowledge a few properties of the modulus of path families that will be important to us. We note that this does not cover all information on modulus of path families, so please consult [24, Chapter II] for further information. First, if every path in the family $\Gamma_1$ is a subpath of the family $\Gamma_2$ , then $M(\Gamma_2) \leq M(\Gamma_1)$ . In other words, shorter paths will result in a larger modulus. To see this in action, let $0 < a < b < \infty$ and the ring domain $A = \{x \in \mathbb{R}^n : a < |x| < b\}$ . Also, set $\Gamma_A$ to be the family of paths joining $\{x : |x| = a\}$ and $\{x : |x| = b\}$ in A. Then, $$M(\Gamma_A) = \omega_{n-1} \left( \log \left( \frac{b}{a} \right) \right)^{1-n},$$ where $\omega_{n-1}$ is the (n-1)-dimensional measure of the unit sphere $S^{n-1}$ . We can also define a more general ring domain.

Definition 2.9. A ring domain is a domain whose complement consists of a bounded and an unbounded component which we denote as $D_0$ and $D_1$ , respectively. The boundary components of the ring are then $\partial D_0$ and $\partial D_1$ . We denote the ring domain by $R(D_0, D_1)$ . When considering ring domains in particular, there is a function due to Väisälä [24] which will be of particular use for estimates.

Definition 2.10. Given r > 0, we let $\Phi_n(r)$ be the set of all rings $A = R(D_0, D_1)$ in $S^n$ , where $D_0, D_1 \subset S^n$ are the complementary components of the ring, with the following properties: - (a) $D_0$ contains the origin and a point a such that |a| = 1. - (b) $D_1$ contains $\infty$ and a point b such that |b| = r. We denote $$\mathcal{H}_n(r) = \inf M(\Gamma_A),$$ <span id="page-7-1"></span>over all rings $A \in \Phi_n(r)$ .

Definition 2.15. Let $f:(X,d_X) \to (Y,d_Y)$ be a homeomorphism. Then f is said to be $\eta$ -quasisymmetric if there exists an increasing homeomorphism $\eta:[0,\infty) \to [0,\infty)$ such that for every distinct triple of points $x,y,z \in X$ , we have $$\frac{d_Y(f(x), f(y))}{d_Y(f(x), f(z))} \le \eta \left(\frac{d_X(x, y)}{d_X(x, z)}\right).$$ There is a weaker version of the above definition. Proving that the weaker version holds for a function can potentially act as a stepping stone to showing it is $\eta$ -quasisymmetric. This is because in certain settings [15, Thm. 10.19], the two definitions do turn out to be equivalent.

Definition 2.16. Let $f:(X,d_X) \to (Y,d_Y)$ be a map. We say f is weakly H-quasisymmetric if there exists $H \geq 1$ such that for every distinct triple $x, y, z \in X$ , $$d_Y(f(x), f(y)) \le Hd_Y(f(x), f(z)),$$ whenever $d_X(x,y) \leq d_X(x,z)$ . However, this version of the weaker definition is not always ideal to work with. The following proposition provides an equivalent definition to weak quasisymmetry.

Definition 2.25. Suppose X, Y, Z are domains in $S^n$ each equipped with conformal metrics, where $Y \subset Z$ and $\mathcal{F} \subset C(X,Y)$ . Then $\mathcal{F}$ is a normal family relative to Z if $\mathcal{F}$ is relatively compact in C(X,Z) and the closure of $\mathcal{F}$ in C(X,Z) is the closure of $\mathcal{F}$ in C(X,Y) together possibly with constant maps into $\partial Y$ , viewing the boundary of Y as a subset of Z. There are then several equivalent definitions of normal families.

Definition 2.26. Let X,Y be subdomains of $S^n$ with conformal metrics, let $M \geq 1$ and $\mathcal{F} \subset \mathcal{Q}_M(X,Y)$ . Then we say $\mathcal{F}$ is a normal family if any (and hence all) of the following equivalent statements hold: - (i) $\mathcal{F}$ is a normal family relative to $S^n$ in the sense of Definition 2.25; - (ii) $\mathcal{F}$ is relatively compact in $C(X, S^n)$ ; - (iii) every sequence $(f_m)$ in $\mathcal{F}$ has subsequence that converges uniformly on compact subsets of X, in the spherical metric, to a limit function $f: X \to S^n$ . The following is one of the main results from [9]. It is incredibly useful in the case where $\mathcal{F} \subset \mathcal{Q}_M(X, S^n)$ , as it relates normality of a family to local uniform $\omega$ -continuity.

Definition 2.28. Let $\mathcal{F} \subset \mathcal{Q}_M(X,\mathbb{R}^n)$ , where X is domain with a distance function $d_X$ arising from a conformal metric and $\mathbb{R}^n$ is equipped with the Euclidean metric. We say $\mathcal{F}$ is finitely normal if every sequence in $\mathcal{F}$ has a subsequence which converges uniformly on compact sets to an element of $\mathcal{Q}_M(X,\mathbb{R}^n)$ . Then, there exists a result similar to that of Theorem 2.27 but for the finitely normal setting. <span id="page-11-2"></span>Corollary 2.29 ([9], Corollary 3.10). Let $X \subset S^n$ be a domain equipped with distance function $d_X$ arising from a conformal metric and let $\mathcal{F} \subset \mathcal{Q}_M(X,\mathbb{R}^n)$ be a family of M-quasiregular mappings. Then $\mathcal{F}$ is finitely normal if and only if $\mathcal{F}$ is locally uniformly $\omega$ -continuous, with $\omega(t) = t^{\alpha}$ , $\alpha = M^{1/(1-n)}$ , and $\{f(x_0) : f \in \mathcal{F}\}$ is bounded for some $x_0 \in X$ .