Geometric Versions of Schwarz's Lemma for Spherically Convex Functions

Theorem 1.1. (Area Schwarz's Lemma for spherically convex functions). Let $f: \mathbb{D} \to \widehat{\mathbb{C}}$ be spherically convex. Then $$A_s f(r\mathbb{D}) \leq A_s(r\mathbb{D})$$ for every $0 < r < 1$ . Moreover, equality holds for some 0 < r < 1 if and only if f is a spherical isometry. Theorem 1.1 raises the problem whether there exists a corresponding lower bound for the ratio (1.3) $$\mathscr{A}_{s}(r) := \frac{A_{s} f(r \mathbb{D})}{A_{s}(r \mathbb{D})}, \quad r \in (0,1).$$ Note that (1.3) is the spherical analog of the euclidean quantity (1.2). Since <span id="page-2-1"></span> $$\lim_{r\to 0+} \mathscr{A}_{s}(r) = f^{\sharp}(0)^{2},$$ a lower bound for $\mathscr{A}_s(r)$ would follow provided one could prove that $\mathscr{A}_s(r)$ is increasing as a function of r.

Theorem 1.2. Let $f: \mathbb{D} \to \widehat{\mathbb{C}}$ be spherically convex. Then $\mathscr{A}_s(r)$ is a strictly increasing function of $r \in (0,1)$ , unless f is a spherical isometry in which case $\mathscr{A}_s(r) \equiv 1$ . Theorem 1.2 is a spherical analog of the previously known monotonicity results for euclidean and hyperbolic area ([1, 5, 12]) mentioned at the beginning.

Corollary 1.1. Let $f: \mathbb{D} \to \widehat{\mathbb{C}}$ be a spherically convex function. Then $$(1.4) A_s f(r\mathbb{D}) \ge A_s(r\mathbb{D}) f^{\sharp}(0)^2 for every 0 < r < 1.$$ Moreover, equality holds in (1.4) for some 0 < r < 1 if and only if f is a spherical isometry. Remark 1.1 (Theorem 1.2 vs. Theorem 1.1). Clearly, $A_s f(r\mathbb{D}) \leq \pi/2$ for any spherically convex function $f: \mathbb{D} \to \widehat{\mathbb{C}}$ , so Theorem 1.2 easily implies <span id="page-2-3"></span> $$\frac{\mathrm{A_s} f(r\mathbb{D})}{\mathrm{A_s}(r\mathbb{D})} = \mathscr{A_s}(r) \leq \limsup_{\rho \to 1-} \mathscr{A_s}(\rho) \leq \lim_{\rho \to 1-} \frac{\pi/2}{\mathrm{A_s}(\rho \, \mathbb{D})} = \lim_{\rho \to 1-} \frac{1+\rho^2}{2\rho} = 1$$ for every 0 < r < 1. In this sense, Theorem 1.1 appears as an easy corollary of Theorem 1.2. However, the proof of Theorem 1.2 we give below depends in an essential way on Theorem 1.1, so the apparently stronger statement of Theorem 1.2 is in fact equivalent to Theorem 1.1. In passing, we note that the proof of Theorem 1.2 leads to another sharp lower bound for the spherical area $A_s f(r\mathbb{D})$ which is more precise than the one provided by the sharp inequality (1.4), but geometrically less pleasing:

Corollary 1.2. Let $f: \mathbb{D} \to \widehat{\mathbb{C}}$ be a spherically convex function. Then $$A_s \, f(r \mathbb{D}) \geq \frac{\pi r^2}{1 + r^2 f^\sharp(0)^2} f^\sharp(0)^2 \quad \text{ for every } 0 < r < 1 \, .$$ Moreover, equality holds for any 0 < r < 1 if f has the form $f(z) = T(\eta z)$ with $T \in \text{Rot}(\widehat{\mathbb{C}})$ and $0 < |\eta| \le 1$ . Our next results deal with the spherical length $$\mathrm{L}_{\mathrm{s}}\,f(r\,\mathbb{T}):=\int\limits_{r\,\mathbb{T}}f^\sharp(z)\,|dz|$$ of the image ^f(rT) of the circle ^r^T under a meromorphic map ^f : ^D ^→ ^Cb. We denote by $$\mathrm{L}_{\mathrm{s}}(r\mathbb{T}) := \int\limits_{r\mathbb{T}} \frac{|dz|}{1+|z|^2} = \frac{2\pi r}{1+r^2}$$ the spherical length of the circle rT. <span id="page-3-1"></span>Theorem 1.3 (Length Schwarz's lemma for spherically convex functions). Let f : ^D ^→ ^C^b be a spherically convex function. Then $$L_s(f(r\mathbb{T})) \ge L_s(r\mathbb{T}) f^{\sharp}(0)$$ for every $0 < r < 1$ . Moreover, equality holds for some 0 < r < 1 if and only if f is a spherical isometry. <span id="page-3-2"></span>Remark 1.2. An upper bound for L^s ^f(rT) for spherically convex functions ^f : ^D ^→ ^C^b is $$\mathrm{L}_{\mathrm{s}} \, f(r \, \mathbb{T}) \leq \frac{2 \pi r}{1 - r^2} f^\sharp(0) \quad \text{ for every } 0 < r < 1 \, .$$ Similar to Corollary [1.2](#page-2-4) there is also a more precise, but geometrically less natural lower bound for spherical length, which follows from Corollary [1.2](#page-2-4) in conjunction with the isoperimetric inequality. <span id="page-3-3"></span>Theorem 1.4. Let f : ^D ^→ ^C^b be spherically convex. Then $$L_s(f(r\mathbb{T})) \geq \frac{2\pi r f^\sharp(0)}{1 + r^2 f^\sharp(0)^2} \quad \text{for every } 0 < r < 1 \,.$$ Moreover, equality holds for any ⁰ ^&lt; ^r ^&lt; ¹ if f has the form f(z) = ^T(ηz) with T ^∈ Rot(Cb) and 0 < |η| ≤ 1. Theorem [1.2](#page-2-2) raises the question whether the ratio (1.5) $$\mathscr{L}_{s}(r) := \frac{L_{s} f(r\mathbb{T})}{L_{s}(r\mathbb{T})}$$ is monotonically increasing as a function of r. Note that [\(1.5\)](#page-3-0) is the spherical analog of the quantity [\(1.1\)](#page-0-1). While we cannot offer a full answer, we shall now show that such a monotonicity result does hold for spherically convex functions ^f : ^D ^→ ^C^b which are centrally normalized: <span id="page-3-0"></span> $$f(z) = \alpha z + a_3 z^3 + ..., \quad z \in \mathbb{D},$$ where $$\alpha = \max_{z \in \mathbb{D}} \left( 1 - |z|^2 \right) f^{\sharp}(z).$$ The important additional assumption is that f ⁰⁰(0) = 0. The notion of central normalization and the insight of its relevance in the study of spherically convex function is due to Mej´ıa and Pommerenke [\[18\]](#page-18-7) building on earlier work of Minda and Wright [\[21\]](#page-18-11), Chuaqui and Osgood [\[6\]](#page-18-12) and Chuaqui, Osgood and Pommerenke [\[7\]](#page-18-13). According to [\[18,](#page-18-7) Theorem 4], for any spherically convex function ^f there is always a unit disk automorphism ^ψ and a rotation ^T ^∈ Rot(Cb) such that T ◦ f ◦ψ is centrally normalized. <span id="page-4-0"></span>Theorem 1.5. Let f : ^D ^→ ^C^b be a centrally normalized spherically convex function. Then Ls(r) is a strictly increasing function of r ∈ (0,1), unless f is a spherical isometry in which case Ls(r) ≡ 1. Theorem [1.5](#page-4-0) is a spherical analog of the previously known monotonicity results for euclidean and hyperbolic length ([\[23,](#page-18-0) [12\]](#page-18-2)). In addition to spherical length and spherical area another important geometric quantity in spherical geometry is the total spherical curvature $$\int\limits_{\gamma} \kappa_s(w,\gamma) \lambda_{\widehat{\mathbb{C}}}(w) |dw|$$ of a curve γ, see Section [2](#page-5-0) and [\[20\]](#page-18-10). Roughly speaking, total spherical curvature measures how much the curve γ diverges from being a spherical geodesic. We consider the ratio $$\Phi_{s}(r) := \frac{\displaystyle\int\limits_{f(r\,\mathbb{T})} \kappa_{s}(w,f(r\,\mathbb{T}))\,|dw|}{\displaystyle\int\limits_{r\,\mathbb{T}} \kappa_{s}(z,r\,\mathbb{T})\,|dz|}.$$ and prove the following monotonicity property. <span id="page-4-2"></span>Theorem 1.6. Let f : ^D ^→ ^C^b be a centrally normalized spherically convex function. Then Φs(r) is a strictly increasing function of r ∈ (0,1), unless f is a spherical isometry in which case Φs(r) ≡ 1. One of the crucial ingredients of the proofs of the above theorems is a basic result from [\[14,](#page-18-4) Theorem 4] which guarantees that a meromorphic univalent function f in D is spherically convex if and only if the auxiliary function (1.6) $$h_f(z) := \text{Re}\left\{1 + \frac{zf''(z)}{f'(z)} - \frac{2zf'(z)\overline{f(z)}}{1 + |f(z)|^2}\right\}$$ has the property that <span id="page-4-1"></span> $$h_f(z) \ge 0$$ for every $z \in \mathbb{D}$ . This characterization of spherical convexity has an elegant geometric interpretation in terms of the spherical curvature κs(f(z), f(rT) of the curve f(rT) at the point f(z), |z| = r, since $$h_f(z) = \kappa_s(f(z), f(r\mathbb{T}))f^{\sharp}(z)|z|,$$ see [\(2.4\)](#page-7-0) below, so a meromorphic univalent function f in D is spherically convex if and only if $$\kappa(f(z), f(r\mathbb{T})) \ge 0$$ for all $|z| = r$ and all $0 < r < 1$ . For further information on spherical convexity and spherically convex functions we refer to Section 2 and also to [10, 14, 15, 16, 18, 20, 26] as well as to the recent work [9], where monotonicity results are proved regarding the elliptic-area-radius of $f(r\mathbb{D})$ and condenser capacity. Other variants of the Schwarz lemma for meromorphic functions can be found e.g. in [8, 24]. The paper is structured in the following way. In Section 2 we recall a number of basic facts about spherical geometry and spherical convexity which are necessary for our investigations, including the spherical Gauss-Bonnet Theorem and the spherical isoperimetric inequality. In Section 3 we study the auxiliary function $h_f$ defined in (1.6) and give a new characterization of spherical convexity as well as establishing a sharp lower bound for the integral means of $h_f$ . A corresponding pointwise sharp lower estimate for $h_f$ has been given by Mejía and Pommerenke in their important work [17] on the Schwarzian derivative for spherically convex functions. While the estimate of Mejía and Pommerenke is valid only for centrally normalized functions, our 'integrated' version does hold for any spherically convex function and possesses a natural geometric significance in terms of total geodesic curvature. The spherical Schwarz-type lemmas, Theorem 1.1 and 1.3 and Remark 1.2, are proved in Section 4. Then attention shifts to monotonicity results for spherically convex functions. In Section 5 we consider spherical area and prove Theorem 1.2 as well as Corollary 1.2 and Theorem 1.4. The monotonicity of spherical length (Theorem 1.5) and of total spherical curvature (Theorem 1.6) for centrally normalized spherically convex functions is established in Section 6. In a final Section 7 we illustrate by examples that spherical convexity is a basic requirement for Theorems 1.2, 1.5 and 1.6 and that central normalization is a necessary hypothesis for Theorem 1.6. <span id="page-5-0"></span>2 SPHERICAL CONVEXITY - GAUSS BONNET FORMULA - ISOPERIMETRIC INEQUALITY Suppose $f:\mathbb{D}\to\widehat{\mathbb{C}}$ is a meromorphic univalent function and $f(\mathbb{D})$ is a hyperbolic domain in $\widehat{\mathbb{C}}$ .

Lemma 2.1. [10, Theorem 1] The spherical density $(1-|z|^2) f^{\sharp}(z)$ is a superharmonic function on $\mathbb{D}$ if and only if $f(\mathbb{D})$ is a spherically convex domain.

Lemma 2.2. [10, p.288] If f is spherically convex on $\mathbb{D}$ , then $(1-|z|^2) f^{\sharp}(z) \leq 1$ for every $z \in \mathbb{D}$ . Equality holds for some $z \in \mathbb{D}$ if and only if f maps $\mathbb{D}$ onto a hemisphere and z is the spherical center of the hemisphere.

Proposition 2.1. [14, Theorem 4] Let $f : \mathbb{D} \to \widehat{\mathbb{C}}$ be a meromorphic univalent function. Then f is spherically convex if and only if <span id="page-5-1"></span> $$h_f(z) = \operatorname{Re}\left\{1 + \frac{zf''(z)}{f'(z)} - 2\frac{zf'(z)\overline{f(z)}}{1 + |f(z)|^2}\right\} \ge 0, \quad z \in \mathbb{D}.$$ Mejía and Pommerenke, see [17, (3.14)], have proved that for any centrally normalized spherically convex function f, (2.1) $$h_f(z) \ge \frac{1 - |z|^2}{1 + |z|^2}.$$ In fact, it is not difficult for the reader to convince himself that equality can hold in (2.1) for some $z \in \mathbb{D}$ if and only if f is a spherical isometry. For a geometric interpretation of spherical convexity, we briefly discuss the notion of spherical curvature. A curve $\gamma$ is said to have spherical curvature $\kappa_s(z, \gamma)$ equal to 0 at any of its points if and only if it is a spherical geodesic. Let $\gamma: z = z(t)$ be a $C^2$ curve on $\widehat{\mathbb{C}}$ with everywhere non-vanishing tangent. The spherical curvature of $\gamma$ at z(t) is <span id="page-6-1"></span> $$\kappa_{s}(z(t),\gamma)\lambda_{\widehat{\mathbb{C}}}(z(t)) = k(z(t),\gamma) - \operatorname{Im}\left\{\frac{2\overline{z(t)}z'(t)}{(1+|z(t)|^{2})|z'(t)|}\right\},\,$$ where $\kappa(z(t), \gamma)$ is the euclidean curvature of $\gamma$ at z(t). It can easily be calculated that the spherical curvature of $r\mathbb{T}$ at a point $z \in r\mathbb{T}$ is equal to (2.2) $$\kappa_s(z, r\mathbb{T}) = \frac{1 - r^2}{r}.$$

Proposition 2.2. [14, Theorem 3] If $\Omega \subset \mathbb{P}$ has $\mathscr{C}^2$ smooth boundary and $\Omega$ is spherically convex, then for all $z \in \partial \Omega$ , $\kappa_s(z, \partial \Omega) \geq 0$ .

Proposition 2.3. [14, Theorem 2] Suppose $f : \mathbb{D} \to \widehat{\mathbb{C}}$ is a meromorphic univalent function and $\gamma : z = z(t)$ is a $\mathscr{C}^2$ curve in $\mathbb{D}$ . Then <span id="page-6-0"></span>(2.3) $$\kappa_{s}(f(z), f \circ \gamma) \left(1 - |z|^{2}\right) f^{\sharp}(z) = \kappa_{h}(z, \gamma) - \left(1 - |z|^{2}\right) \operatorname{Im} \left\{ \left(2 \frac{\overline{z}}{1 - |z|^{2}} - \frac{f''(z)}{f'(z)} + \frac{2f'(z)\overline{f(z)}}{1 + |f(z)|^{2}}\right) \frac{z'(t)}{|z'(t)|} \right\},$$ where $\kappa_h$ denotes the hyperbolic curvature and $$\kappa_h(z, \gamma) = \left(1 - |z|^2\right) \kappa(z, \gamma) + 2\operatorname{Im}\left\{\frac{\overline{z(t)}z'(t)}{|z'(t)|}\right\}.$$ Let $f: \mathbb{D} \to \widehat{\mathbb{C}}$ be a meromorphic univalent function. For the definition of the function $\Phi_s(r)$ , as stated in the Introduction, we will need the spherical curvature of $f(r\mathbb{T})$ . Therefore, according to (2.3) $$\kappa_{s}(f(z), f(r\mathbb{T})) (1 - r^{2}) f^{\sharp}(z) = \frac{1 + r^{2}}{r} - \frac{1 - r^{2}}{r} \operatorname{Im} \left\{ i \left[ 2 \frac{r^{2}}{1 - r^{2}} - \frac{z f''(z)}{f'(z)} + \frac{2z f'(z) \overline{f(z)}}{1 + |f(z)|^{2}} \right] \right\} \\ = \frac{1 - r^{2}}{r} + \frac{1 - r^{2}}{r} \operatorname{Re} \left\{ \frac{z f''(z)}{f'(z)} - \frac{2z f'(z) \overline{f(z)}}{1 + |f(z)|^{2}} \right\} \\ = \frac{1 - r^{2}}{r} h_{f}(z)$$ for $z = re^{it}, r \in (0,1), t \in [0,2\pi]$ and $\kappa(z,\gamma) = \frac{1}{r}$ . Hence <span id="page-7-0"></span>(2.4) $$\kappa_{s}(f(z), f(r\mathbb{T})) = \frac{h_{f}(z)}{|z|f^{\sharp}(z)},$$ where $z = re^{it}$ . The total spherical curvature is a geometric quantity that measures how much a curve diverges from being spherically convex. From (2.2), the total spherical curvature of $r\mathbb{T}$ is equal to <span id="page-7-4"></span>(2.5) $$\int_{r\mathbb{T}} \kappa_s(z, r\mathbb{T}) \lambda_{\widehat{\mathbb{C}}}(z) |dz| = 2\pi \frac{1 - r^2}{1 + r^2}$$ and from (2.4), the total spherical curvature of $f(r\mathbb{T})$ is <span id="page-7-5"></span>(2.6) $$\int_{f(r\mathbb{T})} \kappa_s(w, f(r\mathbb{T})\lambda_{\widehat{\mathbb{C}}}(w)|dw| = \int_{r\mathbb{T}} \kappa_s(f(z), f(r\mathbb{T}))f^{\sharp}(z)|dz| = \int_{(2.4)}^{2\pi} \int_{0}^{2\pi} h_f(re^{it})dt.$$ For more information on spherical convexity and spherical curvature, the reader may refer to [14, 15, 20]. In the proof of Theorem 1.6, we will use the Gauss-Bonnet formula in the following form, see [25, Theorem 6.5]. Let M be an oriented two-dimensional Riemannian manifold with Gaussian curvature K and volume element dA. Let $N \subset M$ be a compact two-dimensional manifold-with-boundary which is diffeomorphic to a subset of $\mathbb{R}^2$ and whose boundary is connected. Let ds be the volume element of $\partial N$ and let $\kappa$ be the signed geodesic curvature of $\partial N$ . Then <span id="page-7-1"></span>(2.7) $$\int_{N} K dA + \int_{\partial N} \kappa ds = 2\pi.$$ The Riemann sphere $\widehat{\mathbb{C}}$ endowed with the spherical metric is a two-dimensional Riemannian manifold of constant Gaussian curvature equal to 4. If $\Omega$ is a hyperbolic domain in $\widehat{\mathbb{C}}$ , the Gauss-Bonnet formula (2.7) takes the form <span id="page-7-2"></span>(2.8) $$4 A_{s}(\Omega) + \int_{\gamma} \kappa_{s}(z, \gamma) \lambda_{\widehat{\mathbb{C}}}(z) |dz| = 2\pi,$$ where $\gamma$ is the boundary of $\Omega$ assuming that it is a smooth, simple and closed curve in $\widehat{\mathbb{C}}$ . Applying the Gauss-Bonnet formula (2.8) to $f(r\mathbb{D})$ viewed as a two-dimensional manifold with boundary on the Riemann surface of f, we obtain <span id="page-7-3"></span>(2.9) $$\int_{0}^{2\pi} h_f(re^{it}) dt = 2\pi - 4 A_s f(r\mathbb{D}),$$ where $A_s f(r\mathbb{D})$ is the spherical area of $f(r\mathbb{D})$ . Last but not least, in order to prove lower bounds for the spherical length, we use the isoperimetric inequality of spherical geometry; see [22]. Suppose D is a simply connected smooth subdomain of $\widehat{\mathbb{C}}$ . Then <span id="page-8-0"></span>(2.10) $$L_{s}(\partial D)^{2} \ge 4\pi A_{s}(D) - 4A_{s}(D)^{2}.$$ <span id="page-8-3"></span>3 THE FUNCTION $h_f$ For our purposes the following characterization of spherically convex functions in terms of the function <span id="page-8-1"></span> $$h_f(z) = \text{Re}\left\{1 + z \frac{f''(z)}{f'(z)} - \frac{2z\overline{f(z)}f'(z)}{1 + |f(z)|^2}\right\}$$ turns out to be useful.

Theorem 3.1. Let $f: \mathbb{D} \to \widehat{\mathbb{C}}$ be a meromorphic univalent function. Then (3.1) $$\Delta h_f(z) = -8f^{\sharp}(z)^2 h_f(z).$$ In particular, f is spherically convex if and only if $h_f$ is superharmonic on $\mathbb{D}$ . In this case, $h_f$ is strictly superharmonic, so $\Delta h_f < 0$ in $\mathbb{D}$ .

Theorem 3.2. Let $f: \mathbb{D} \to \widehat{\mathbb{C}}$ be spherically convex. Then for any $r \in (0,1)$ <span id="page-9-0"></span>(3.2) $$\frac{1}{2\pi} \int_{0}^{2\pi} h_f(re^{it}) dt \ge \frac{1 - r^2}{1 + r^2}.$$ For fixed $r \in (0,1)$ equality holds in (3.2) if and only if f is a spherical isometry. Theorem 3.2 is an integrated version of the Mejía–Pommerenke inequality (2.1), but with the additional benefit that we do not need to assume central normalization. The estimate (3.2) has a natural geometric interpretation by observing that the integral expression is precisely the normalized total spherical curvature of $f(r\mathbb{T})$ , while the right-hand side is the normalized total spherical curvature of the circle $r\mathbb{T}$ , see Section 2.

Proposition 3.1. Let $f: \mathbb{D} \to \widehat{\mathbb{C}}$ be a meromorphic univalent function. Then (3.3) $$\int_{0}^{2\pi} f^{\sharp}(re^{it})^{2} dt = \frac{2}{r^{2}} \iint_{r \mid \mid \mid} h_{f}(z) f^{\sharp}(z)^{2} dA(z)$$ for any $r \in (0, 1)$ .