On Zalcman's and Bieberbach conjectures

Theorem 1. For any function $f \in S$ and any $n \geq 3$ , we have the sharp bound $|J_{1,2}(f)| \leq 1$ , with equalities $$\max_{S} |J_1(f)| = \max_{S} |J_2(f)| = 1 \tag{6}$$ only for the Koebe function (2) and its rotations $e^{-i\alpha}\kappa_{\theta}(e^{i\alpha}z)$ . The proof of this theorem leads to the bound $J_{1,2}(f) \leq 1$ for all $f \in S$ , with the equality $J_{1,2}(f_0) = 1$ for any extremal function maximizing $J_{1,2}$ . This yields by (5) that either $\max_S |J_1(f_0)| = 1$ and $\max_S |J_2(f_0)| \leq 1$ , or $\max_S |J_1(f_0)| \leq 1$ and $\max_S |J_2(f_0)| = 1$ , or both equalities (6). In either case the extremality is possible only for $f_0 = \kappa_\theta$ (and then both these maxima must be equal to 1). 1.3. On extremality of Koebe's function. The spaces $\mathcal{T}_m$ possess an important property inherited from the classes $\Sigma$ and $\widehat{\Sigma}(1)$ inherited from the class $\Sigma$ and preserved under connection (1). This is a circular symmetry of maps $F_{\tau,\sigma}$ , since for any $F \in \Sigma$ , all functions $e^{-i\alpha}F(e^{i\alpha}z)$ also belong to this class. Theorem 1 intrinsically relates to the general result on extremality of Koebe's function established in [22] concerning the arbitrary rotationally homogeneous polynomial functionals $J(f) = J(a_{m_1}, \ldots, a_{m_s})$ on the class S (with $2 < a_{m_1} < \cdots < a_{m_s} < \infty$ ). The relations between the coefficients of $f(z) = z + a_2 z^2 + \cdots \in S$ and of their inversions $F(z) = 1/f(1/z) = z + b_0 + b_1 z^{-1} + \ldots$ , which belong to the class $\Sigma$ of univalent $\widehat{\mathbb{C}}$ -holomorphic functions on the disk $\mathbb{D}^* = \{z \in \widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\} : |z| > 1\}$ with such expansions, transform J(f) to a functional $\widetilde{J}(F)$ on $\Sigma$ , which is called to be associated with J (see below the relations (20)). Denote by $\Sigma_{\rm af}$ the subcollection in $\Sigma$ formed by functions $$F_{b_0,b_1:t}(z) = z + b_0t + b_1t^2z^{-1}$$ with $$|b_0| \le 2$$ , $|b_1| \le 1$ , $|t| \le 1$ . These functions have the affine extensions $\widehat{F}_{b_0,b_1;t}(z) = z + b_0t + b_1t^2\overline{z}$ onto the unit disk $\mathbb{D} = \{|z| < 1\}$ (with constant dilatations $b_1t^2$ ). The indicated result states:

Theorem 2. The Koebe function $\kappa_{\theta}(z)$ is (a unique) extremal of a rotationally homogeneous coefficient functional J(f) if and only if its associated functional $\widetilde{J}(F)$ satisfies $$\max_{\Sigma} |\widetilde{J}(F)| = \max_{f \in S} |\widetilde{J}(F_f)| = \sup_{\Sigma_{\mathrm{af}}} |\widetilde{J}(F_{b_0, b_1; t})|.$$ In other words, the maximal value of the associate to J functional $\widetilde{J}$ on $\Sigma$ must be attained on the distinguished subset $\Sigma_{\mathrm{af}}$ of functions $F \in \Sigma$ admitting the affine extensions to the disk $\mathbb{D}$ . This theorem shows that actually the collection of coefficient functionals maximized by Koebe's function is rather sparse. Giving rise to possible generalizations of the Bieberbach and Zalcman conjectures, it simultaneously describes the admissible extent of this direction. Note that this theorem involves only the weak rotational homogeneity of J(f).

Lemma 1. For any Beltrami coefficient $\mu \in \text{Belt}(\mathbb{D}^*)_1$ and any $\theta_0 \in [0, 2\pi]$ , there exists a point $z_0 = e^{i\alpha}$ located on $\mathbb{S}^1$ so that $|e^{i\theta_0} - e^{i\alpha}| < 1$ and such that for any $\theta$ satisfying $|e^{i\theta} - e^{i\alpha}| < 1$ the equation $\partial_{\overline{z}} w = \mu(z) \partial_z w$ has a unique homeomorphic solution $w = w^{\mu}(z)$ , which is holomorphic on the unit disk $\mathbb{D}$ and satisfies $$w(0) = 0, \quad w'(0) = e^{i\theta}, \quad w(z_0) = z_0.$$ (7) Hence, $w^{\mu}(z)$ is conformal and does not have a pole in $\mathbb{D}$ (so $w^{\mu}(z_) = \infty$ at some point $z$ with $|z_*| \geq 1$ ). In particular, this lemma allows one to define the Teichmüller spaces using the quasiconformally extendible univalent functions w(z) in the unit disk $\mathbb{D}$ normalized by $$w(0) = 0, \quad w'(0) = 1, \quad w(1) = 1$$ (8) and with more general normalization $$w(0) = 0$$ , $w'(0) = e^{i\theta}$ , $w(1) = 1$ . All such functions are holomorphic in the disk $\mathbb{D}$ . As a simple consequence of this lemma, we have, passing to rotations $w(z) \mapsto e^{i\alpha} w(e^{-i\alpha}z)$ , the following

Lemma 2. For any $\mu \in \text{Belt}(\mathbb{D}^*)_1$ , any given value $\theta \in (-\pi, \pi]$ and any point $z_0$ of the unit circle $\mathbf{S}^1 = \{|z| = 1\}$ , there exists a unique homeomorphic solution $w = w^{\mu}(z)$ of the equation $\partial_{\overline{z}}w = \mu(z)\partial_z w$ on $\widehat{\mathbb{C}}$ such that $$w(0) = 0, \quad w'(0) = 1, \quad w(z_0) = e^{i\theta}.$$ (9) This solution is holomorphic on the unit disk $\mathbb{D}$ , and hence, $w(z_) = \infty$ at some point $z$ with $|z_*| \geq 1$ . Note that for $\mu(z) = 0$ (almost everywhere on $\mathbb{D}^*$ ) the corresponding solution $w^{\mu}(z)$ with w(0) = 0, $w'(0) = e^{i\theta}$ , w(1) = 1 is the elliptic Móbius map $$w = \frac{e^{-i\theta}z}{(e^{-i\theta} - 1)z + 1}$$ (with the fixed points 0 and 1); it equals the identity map if $\theta = 0$ . 2.3. The points of Teichmüller space $\mathbf{T}_1 = \mathrm{Teich}(\mathbb{D}_)$ of the punctured disk $\mathbb{D}_ = \mathbb{D} \setminus \{0\}$ are the classes $[\mu]_{\mathbf{T}_1}$ of $\mathbf{T}_1$ -equivalent Beltrami coefficients $\mu \in \mathrm{Belt}(\mathbb{D})_1$ , which means that the corresponding quasiconformal automorphisms $w^{\mu}$ of the unit disk coincide on both boundary components of $\mathbb{D}_*$ (the unit circle $\mathbb{S}^1 = \{|z| = 1\}$ and the puncture z = 0) and are homotopic on $\mathbb{D} \setminus \{0\}$ . This space can be endowed with a canonical complex structure of a complex Banach manifold and embedded into $\mathbf{T}$ using uniformization. Namely, the disk $\mathbb{D}$ is conformally equivalent to the factor $\mathbb{D}/\Gamma$ , where $\Gamma$ is a cyclic parabolic Fuchsian group acting discontinuously on $\mathbb{D}$ and $\mathbb{D}$ . The functions $\mu \in L_{\infty}(\mathbb{D})$ are lifted to $\mathbb{D}$ as the Beltrami (-1,1)-measurable forms $\widetilde{\mu}d\overline{z}/dz$ in $\mathbb{D}$ with respect to $\Gamma$ , i.e., via $(\widetilde{\mu} \circ \gamma)\overline{\gamma'}/\gamma' = \widetilde{\mu}, \ \gamma \in \Gamma$ , forming the Banach space $L_{\infty}(\mathbb{D},\Gamma)$ . We extend these $\widetilde{\mu}$ by zero to $\mathbb{D}$ and consider the unit ball $\operatorname{Belt}(\mathbb{D},\Gamma)_1$ of $L_{\infty}(\mathbb{D},\Gamma)$ . Then the corresponding Schwarzians $S_{w\widetilde{\mu}|\mathbb{D}}$ belong to $\mathbf{T}$ . Moreover, $\mathbf{T}_1$ is canonically isomorphic to the subspace $\mathbf{T}(\Gamma) = \mathbf{T} \cap \mathbf{B}(\Gamma)$ , where $\mathbf{B}(\Gamma)$ consists of elements $\varphi \in \mathbf{B}$ satisfying $(\varphi \circ \gamma)(\gamma')^2 = \varphi$ in $\mathbb{D}^*$ for all $\gamma \in \Gamma$ . Due to the Bers isomorphism theorem, the space $T_1$ is biholomorphically isomorphic to the Bers fiber space $$\mathcal{F}(\mathbf{T}) = \{ (\phi_{\mathbf{T}}(\mu), z) \in \mathbf{T} \times \mathbb{C} : \mu \in \text{Belt}(\mathbb{D})_1, z \in w^{\mu}(\mathbb{D}) \}$$ over the universal space T with holomorphic projection $\pi(\psi, z) = \psi$ (see [3]). This fiber space is a bounded hyperbolic domain in $\mathbf{B} \times \mathbb{C}$ and represents the collection of domains $D_{\mu} = w^{\mu}(\mathbb{D})$ as a holomorphic family over the space $\mathbf{T}$ . For every $z \in \mathbb{D}$ , its orbit $w^{\mu}(z)$ in $\mathbf{T}_1$ is a holomorphic curve over $\mathbf{T}$ . The indicated isomorphism between $\mathbf{T}_1$ and $\mathcal{F}(\mathbf{T})$ is induced by the inclusion map $j: \mathbb{D}_* \hookrightarrow \mathbb{D}$ forgetting the puncture at the origin via $$\mu \mapsto (S_{w^{\mu_1}}, w^{\mu_1}(0)) \quad \text{with} \quad \mu_1 = j_* \mu := (\mu \circ j_0) \overline{j_0'} / j_0',$$ (10) where $j_0$ is the lift of j to $\mathbb{D}$ . The Bers theorem is valid for Teichmüller spaces $\mathbf{T}(X_0 \setminus \{x_0\})$ of all punctured hyperbolic Riemann surfaces $X_0 \setminus \{x_0\}$ and implies that $\mathbf{T}(X_0 \setminus \{x_0\})$ is biholomorphically isomorphic to the Bers fiber space Fib $(\mathbf{T}(X_0))$ over $\mathbf{T}(X_0)$ . 2.4. The spaces $\mathbf{T}$ and $\mathbf{T}_1$ can be weakly (in the topology generated by the spherical metric on $\widehat{\mathbb{C}}$ ) approximate by finite dimensional Teichmüller spaces $\mathbf{T}(0,n)$ of punctured spheres (Riemann surfaces of genus zero) $$X_{\mathbf{z}} = \widehat{\mathbb{C}} \setminus \{0, 1, z_1 \dots, z_{n-3}, \infty\}$$ defined by ordered n-tuples $\mathbf{z} = (0, 1, z_1, \dots, z_{n-3}, \infty), n > 4$ with distinct $z_j \in \mathbb{C} \setminus \{0, 1\}$ (the details see, e.g., in [19]). Fix a collection $\mathbf{z}^0 = (0, 1, z_1^0, \dots, z_{n-3}^0, \infty)$ with $z_j^0 \in \mathbb{S}^1$ defining the base point $X_{\mathbf{z}^0}$ of the space $\mathbf{T}(0, n) = \mathbf{T}(X_{\mathbf{z}^0})$ . Its points are the equivalence classes $[\mu]$ of Beltrami coefficients from the ball $\mathrm{Belt}(\mathbb{C})_1 = \{\mu \in L_\infty(\mathbb{C}) : \|\mu\|_\infty < 1\}$ under the relation: $\mu_1 \sim \mu_2$ , if the corresponding quasiconformal homeomorphisms $w^{\mu_1}, w^{\mu_2} : X_{\mathbf{a}^0} \to X_{\mathbf{a}}$ are homotopic on $X_{\mathbf{a}^0}$ (and hence coincide in the points $0, 1, z_1^0, \dots, z_{n-3}^0, \infty$ ). This models $\mathbf{T}(0, n)$ as the quotient space $\mathbf{T}(0, n) = \mathrm{Belt}(\mathbb{C})_1/\infty$ with complex Banach structure of dimension n-3 inherited from the ball $\mathrm{Belt}(\mathbb{C})_1$ . Another canonical model of the space $\mathbf{T}(0,n) = \mathbf{T}(X_{\mathbf{z}^0})$ is obtained again using the uniformization. The surface $X_{\mathbf{z}^0}$ is conformally equivalent to the quotient space $U/\Gamma_0$ , where $\Gamma_0$ is a torsion free Fuchsian group of the first kind acting discontinuously on $\mathbb{D} \cup \mathbb{D}^*$ . The functions $\mu \in L_{\infty}(X_{\mathbf{z}^0})$ are lifted to $\mathbb{D}$ as the Beltrami (-1,1)-measurable forms $\widetilde{\mu}d\overline{z}/dz$ in $\mathbb{D}$ with respect to $\Gamma_0$ which satisfy $$(\widetilde{\mu} \circ \gamma)\overline{\gamma'}/\gamma' = \widetilde{\mu}, \ \gamma \in \Gamma_0,$$ and form the Banach space $L_{\infty}(\mathbb{D}, \Gamma_0)$ . After extending these $\widetilde{\mu}$ by zero to $\mathbb{D}$ , the Schwarzians $S_{w^{\widetilde{\mu}}|\mathbb{D}}$ for $\|\widetilde{\mu}\|_{\infty} < 1$ belong to $\mathbf{T}$ and form its subspace regarded as the Teichmüller space $\mathbf{T}(\Gamma_0)$ of the group $\Gamma_0$ . It is canonically isomorphic to the space $\mathbf{T}(X_{\mathbf{z}^0})$ , and moreover, $$\mathbf{T}(\Gamma_0) = \mathbf{T} \cap \mathbf{B}(\Gamma_0),$$ where $\mathbf{B}(\Gamma_0)$ is an (n-3)-dimensional subspace of $\mathbf{B}$ which consists of elements $\varphi \in \mathbf{B}$ satisfying $(\varphi \circ \gamma)(\gamma')^2 = \varphi$ for all $\gamma \in \Gamma_0$ (holomorphic $\Gamma_0$ -automorphic forms of degree -4); see, e.g. [25]. This space has has the same elements as the space $A_1(\mathbb{D}^*, \Gamma_0)$ of integrable holomorphic forms of degree -4. This leads to the representation of the space $\mathbf{T}(X_{\mathbf{z}^0})$ as a bounded domain in the complex Euclidean space $\mathbb{C}^{n-3}$ . Any Teichmüller space is a complete metric space with intrinsic Teichmüller metric defined by quasiconformal maps. By the Royden-Gardiner theorem, this metric equals the hyperbolic Kobayashi metric determined by the complex structure (see, e.g., [6], [8], [29]).

Lemma 3. [14] (a) Each homotopy map $F_t$ admits k-quasiconformal extension to the whole sphere $\widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$ with $k \leq |t|^2$ . The bound $k(F_t) \leq |t|^2$ is sharp and occurs only for the maps $$F_{b_0,b_1;1}(z) = z + b_0 + b_1 z^{-1}, \quad |b_1| = 1,$$ whose homotopy maps $$F_{b_0,b_1;t}(z) = z + b_0 t + b_1 t^2 z^{-1}$$ (12) have the affine extensions $\widehat{F}_{b_0,b_1;t}(z) = z + b_0t + b_1t^2\overline{z}$ onto $\mathbb{D}$ . (b) If $F(z) = z + b_0 + b_p z^{-p} + b_{p+1} z^{-(p+1)} + \dots$ ( $b_p \neq 0$ ) for some integer p > 1, then the minimal dilatation of extensions is estimated by $k(F_t) \leq |t|^{p+1}$ ; this bound also is sharp. This lemma is a special case of the dynamical $r^2$ -property of domains and univalent functions investigated in [14], [24]). If $F \in \Sigma_Q$ admits a k-quasiconformal extension, then the best bound for the dilatation of its homotopy $F_t$ much stronger. This bound will not be applied here. Note also that, due to Strebel's frame mapping condition [31], the extremal extensions $F_t$ of any homotopy functions $F_t$ with |t| < 1 is of Teichmüller type, i.e., with the Beltrami coefficient of the form $$\mu_{\widehat{F}_t}(z) = \tau(t)|\psi(z)|/\psi(z),$$ where $\psi$ is a holomorphic function from $L_1(\mathbb{D})$ (and unique). 3.2. It suffices for our goals to consider the functions $f \in S$ with $$b_1 = a_2^2 - a_3 \neq 0;$$ this assumption is equivalent to $S_f(0) = \lim_{z \to \infty} z^4 S_{F_f}(z) \neq 0$ . Such functions form a dense subset of S, and their Schwarzians form a dense subset of the space $\mathbf{T}$ . We divide every homotopy function $F_t$ of $F = F_f$ into two parts $$F_t(z) = z + b_0 t + b_1 t^2 z^{-1} + b_2 t^3 z^{-2} + \dots = F_{b_0, b_1; t}(z) + h(z, t),$$ where $F_{b_0,b_1;t}$ is the map (12) with $b_0$ , $b_1$ coming from F. For sufficiently small |t|, the remainder h is estimated by $h(z,t) = O(t^3)$ uniformly in z for all $|z| \ge 1$ . Then the Schwarzian derivatives of $F_t$ and $F_{b_0,b_1;t}$ are related by $$S_{F_t}(z) = S_{F_{b_0,b_1;t}}(z) + \omega(z,t),$$ where the remainder $\omega$ is uniquely determined by the chain rule $$S_{w_1 \circ w}(z) = (S_{w_1} \circ w)(w')^2(z) + S_w(z),$$ and is estimated in the norm of B by and is estimated in the norm of B by $$\|\omega(\cdot,t)\|_{\mathbf{B}} = O(t^3), \quad t \to 0; \tag{13}$$ this estimate is uniform for $|t| < t_0$ (cf., e.g. [2], [13]). All functions $F_{b_0,b_1;t}$ with $$|b_0| < 2, |b_1| < 1, |t| < 1.$$ (14) are univalent on the disk $\mathbb{D}^*$ (but can vanish there) and, if |t| < 1, have the affine extensions onto $\mathbb{D}$ . For such functions, their homotopy disk $\mathbb{D}(F) = \{F_t\}$ coincides with the extremal disk $\mathbb{D}(\psi) = \{t\mu_0 : t \in \mathbb{D}\} \subset \text{Belt}(\mathbb{D})_1$ ; hence, the action of functional $\widetilde{Z}_n$ on external disks of functions $F_{b_0,b_1;t}$ is rotationally symmetric with respect to $t \in \mathbb{D}$ . We call the values $b_0$ and $b_1$ admissible if they are the initial coefficients of some function from $\Sigma_Q$ (these values satisfy (14)). The collection of all such $F_t$ with |t| < 1 will be denoted by $\Sigma_{\rm af}$ . 3.3. Two generalizations of the Gaussian curvature. The next geometric results of this section are needed for the Zalcman's part of Theorem 1. It relies on the properties of subharmonic conformal metrics $\lambda(z)|dz|$ on the disk (with $\lambda(z) \geq 0$ ), whose curvature is at most -4 in the generalized sense introduced by Ahlfors and Royden (see [1], [12], [29]). Recall that the Gaussian curvature of a $C^2$ -smooth metric $\lambda > 0$ is defined by $$\kappa_{\lambda} = -\frac{\mathbb{D}\log\lambda}{\lambda^2},$$ where $\mathbb{D}$ means the Laplacian $4\partial \overline{\partial}$ . A metric $\lambda(z)|dz|$ in a domain G on $\mathbb C$ (or on a Riemann surface) has curvature less than or equal to K in the supporting sense if for each K' > K and each $z_0$ with $\lambda(z_0) > 0$ , there is a $C^2$ -smooth supporting metric $\widehat{\lambda}$ for $\lambda$ at $z_0$ (i.e., such that $\widehat{\lambda}(z_0) = \lambda(z_0)$ and $\widehat{\lambda}(z) \leq \lambda(z)$ in a neighborhood of $z_0$ ) with $\kappa_{\widehat{\lambda}}(z_0) \leq K'$ , or equivalently, $$\mathbb{D}\log\lambda \ge -K\lambda^2. \tag{15}$$ A metric $\lambda$ has curvature at most K in the potential sense at $z_0$ if there is a disk U about $z_0$ in which the function $$\log \lambda + K \operatorname{Pot}_U(\lambda^2),$$ where $Pot_U$ denotes the logarithmic potential $$\operatorname{Pot}_{U} h = \frac{1}{2\pi} \iint_{U} h(\zeta) \log |\zeta - z| d\xi d\eta \quad (\zeta = \xi + i\eta),$$ is subharmonic. Since $\mathbb{D} \operatorname{Pot}_U h = h$ (in the sense of distributions), one can replace U by any open subset $V \subset U$ , because the function $\operatorname{Pot}_U(\lambda^2) - \operatorname{Pot}_V(\lambda^2)$ is harmonic on U. The inequality (15) holds for the generic subharmonic metrics also in the sense of distributions. Note also that the condition of having curvature at most -K in the potential sense is invariant under conformal maps.

Lemma 4. [29] If a conformal metric has curvature at most K in the supporting sense, then it has curvature at most K in the potential sense. The following lemma concerns the circularly symmetric metrics which are the functions of r = |z|.

Lemma 5. [18], [29] Let $\lambda(|t|)d|t|$ be a circularly symmetric subharmonic metric on $\mathbb{D}$ such that $$\lambda(r) = mcr^{m-1} + O(r^m)$$ as $r \to 0$ with $0 < c \le 1$ $(m = 1, 2, ...)$ (16) and this metric has curvature at most -4 in the potential sense. Then $$\lambda(r) \ge \frac{mcr^{m-1}}{1 - c^2r^{2m}}.\tag{17}$$ In the case m=1 (with $c \neq 0$ ) applied in [29], the estimate (17) is reduced to $\lambda(r) \geq c/(1-c^2r^2)$ with c=(0), and right-hand side defines a supporting conformal metric for $\lambda$ at the origin with constant Gaussian curvature -4 on the whole disk $\mathbb{D}$ . The dominant of metrics subject to (16) is given by the following

Lemma 6. Let $\lambda(t)|dt|$ be a continuous conformal metric on the disk $\mathbb{D}$ with growth (17) near the origin and having the curvature -4 in the supporting sense at its noncritical points. Then $$\lambda(t) \le \lambda_m(t) = \frac{m|t|^{m-1}}{1 - 2t^{2m}}$$ for all $t \in \mathbb{D}$ . 3.3. Restoration of functional $Z_n(f)$ by its infinitesimal form. We also mention two important features of Zalcman's functional. Pass to the normalized functional $Z_n^0 = Z_n/\max_S |Z_n(f)|$ mapping S onto the unit disk, and consider its action on infinite holomorphic families $\mathcal{F}_S = \{f_t(z) = f(z,t)\} \subset S$ and on $\Sigma_{\text{af}}$ with $F(z,\cdot) = 1/f(1/z,\cdot)$ , $t \in \mathbb{D}$ . Using the relations between the coefficients $a_n(t)$ of f(z,t) and the corresponding coefficients $b_j(t)$ of $F_f(z,t)$ , we represent $Z_n^0$ as a polynomial functional on $\Sigma$ , $$Z_n^0(f) = \widetilde{Z}_n^0(F_f) = \widetilde{Z}_n^0(b_0, b_1, \dots, b_{2n-3}).$$ The given holomorphic families determine the sequences of holomorphic maps $$g_m(t) = \widetilde{Z}_n(F_m(\cdot, t)) : \mathbb{D} \to \mathbb{C}, \quad m = 1, 2, \dots (F_m \in \Sigma_{af}),$$ and their upper envelope $$\widehat{g} = \sup_{m} |g_m(t)| : \mathbb{D} \to \mathbb{D}$$ followed by upper semicontinuous regularization $\widehat{g}(t) = \limsup_{t' \to t} g_m(t')$ presents a logarithmically subharmonic function on the unit disk. The maps $g_m$ pull back the hyperbolic metric of this disk $\lambda_{\mathbb{D}}(z)|dz|$ generating on $\mathbb{D}$ the logarithmically subharmonic conformal metrics $ds = \lambda_{g_m}(t)|dz|$ with $$\lambda_{g_m}(t) = g_m^* \lambda_{\mathbb{D}}(t) = \frac{|g_m'(t)|}{1 - |g_m(t)|^2}$$ of Gaussian curvature -4 at noncritical points of $g_m$ . Passing to the upper envelope $$\lambda_{Z_n^0}(z) = \sup_{m} \lambda_{g_m}(z) \tag{18}$$ and its upper semicontinuous regularization, one obtains a logarithmically subharmonic metric on $\mathbb{D}$ , whose curvature is less than or equal to -4 in both supporting and potential senses (cf. [18]). We shall apply the results on the curvatures indicated above to the values of $\widetilde{Z}_n$ on the homotopy disks; thus the derivatives must be understand as distributional, because generically these disks have the critical points. The following lemma from [18] provides that on extremal Teichmüller disks the functional $Z_n^0$ can be reconstructed from its matric $\lambda_{Z_n^0}$ .

Lemma 7. On any extremal Teichmüller disk $\mathbb{D}(\psi_0) = \{t|\psi_0|/\psi_0: |t| < 1\} \subset \text{Belt}(\mathbb{D})_1$ , we have the equality $$\tanh^{-1}\left[\widetilde{Z}_n^0(F^{r|\psi_0|/\psi_0})\right] = \int_0^r \lambda_{\widetilde{Z}_n^0}(t)dt$$ for each r < 1. In particular, this lemma implies the following result which is crucial for the proof of Zalcman's part of Theorem 1.

Lemma 8. [18] For any $$F(z) = z + b_0 + b_1 z^{-1} + \dots \in \Sigma_Q$$ , we have the lower bound $$|\widetilde{Z}_n(F)| \ge \max_{b_0} |\widetilde{Z}_n(F_{b_0,b_1;1})|, \tag{19}$$ where the maximum is taken over the set of admissible $b_0$ . 3.4. Generalization. The above lemmas, especially Lemma 7 on restoration of $Z_n$ by its infinitesimal form are also valid for functionals considered by Theorem 2, for example, $J(f) = a_n$ with n > 2. This lemma deals with the extremal Teichmüller disks in the sense of T-equivalence, i.e., embedded in the universal space T. The second coefficient $a_2$ is somewhat specific. It is a part of the initial conditions w(0) = 0, w'(0) = 1, $w''(0) = 2a_2$ defining the unique solution of the Schwarzian differential equation $S_w = \varphi$ with given $\varphi$ and intrinsically relates to quadratic differential $z^{-3}dz^2$ on $\widehat{\mathbb{C}}$ having a simple pole at infinity. This differential expresses a quasiconformal variation $a_2(w^{\mu})$ via $$a_2(w^{\mu}) = -\frac{1}{\pi} \iint_{\mathbb{D}^*} \frac{\mu(\zeta)}{\zeta^3} d\xi d\eta + O(\|\mu\|_{\infty}^2), \quad \|\mu\|_{\infty} \to 0.$$ The indicated differential is holomorphic only on the punctured disk $\mathbb{D}^* \setminus \{\infty\}$ , and the corresponding Teichmüller disk $\{tz^3/|z|^3\}$ is extremal in the space $\mathbf{T}_1$ , but not in $\mathbf{T}$ . The extremal dilatation along this disk is estimated by the hyperbolic distance of this punctured disk, which is greater that the distance on $\mathbb{D}^*$ . 3.5. Special quasiconformal deformations. The following variational lemma is a special case of the general quasiconformal deformations constructed in [13].

Lemma 9. Let D be a simply connected domain on the Riemann sphere $\widehat{\mathbb{C}}$ . Assume that there are a set E of positive two-dimensional Lebesgue measure and a finite number of points $z_1, z_2, ..., z_m$ distinguished in D. Let $\alpha_1, \alpha_2, ..., \alpha_m$ be non-negative integers assigned to $z_1, z_2, ..., z_m$ , respectively, so that $\alpha_j = 0$ if $z_j \in E$ . Then, for a sufficiently small $\varepsilon_0 > 0$ and $\varepsilon \in (0, \varepsilon_0)$ , and for any given collection of numbers $w_{sj}$ , $s = 0, 1, ..., \alpha_j$ , j = 1, 2, ..., m which satisfy the conditions $w_{0j} \in D$ , $$|w_{0j} - z_j| \le \varepsilon$$ , $|w_{1j} - 1| \le \varepsilon$ , $|w_{sj}| \le \varepsilon$ $(s = 0, 1, \dots a_j, j = 1, \dots, m)$ , there exists a quasiconformal automorphism h of D which is conformal on $D \setminus E$ and satisfies $$h^{(s)}(z_j) = w_{sj}$$ for all $s = 0, 1, ..., \alpha_j, \ j = 1, ..., m$ . Moreover, the Beltrami coefficient $\mu_h(z) = \partial_{\overline{z}}h/\partial_z h$ of h on E satisfies $\|\mu_h\|_{\infty} \leq M\varepsilon$ . The constants $\varepsilon_0$ and M depend only upon the sets D, E and the vectors $(z_1, ..., z_m)$ and $(\alpha_1, ..., \alpha_m)$ . If the boundary $\partial D$ is Jordan or is $C^{l+\alpha}$ -smooth, where $0 < \alpha < 1$ and $l \ge 1$ , we can also take $z_j \in \partial D$ with $\alpha_j = 0$ or $\alpha_j \le l$ , respectively.

Lemma 10. Every function $u_{\theta}(t)$ defiend by (22) with a fixed $\theta \in [-\pi, \pi]$ is logarithmically subharmonic in some domains $D_{\theta}$ located in the disk $\mathbb{D}_4 = \{|t| < 4\}$ . Proof. Fix $\theta \in [-\pi, \pi]$ and, using the maps $F^{\mu} \in \Sigma_{Q,\theta}$ , apply a weak approximation of the underlying space T (and simultaneously of the space T₁) by finite dimensional Teichmüller spaces of the punctured spheres in the topology of locally uniform convergence on $\mathbb{C}$ . Take the set of points $$E = \{e^{\pi si/2^n}, \ s = 0, 1, \dots, 2^{n+1} - 1; \ n = 1, 2, \dots\}$$ ^&lt;sup>1</sup>Recall that one has to deal with quadratic differentials $\varphi(z)dz^2$ to insure the needed behaviour of the Schwarzian derivatives $\varphi(h(z))h'(z)^2 = \varphi(z)$ under conformal changes h of variables. This involves the equivalence classes of $\varphi \in \mathbf{B}$ so that $\varphi$ and $\varphi_1$ are equivalent if $\varphi_1(z) = \epsilon_1 \varphi(\epsilon_2 z)$ for some constants $\epsilon_1$ , $\epsilon_2$ with modulus 1. Such equivalence preserves $\mathbf{B}$ norm and moduli of coefficients. The homogeneity of $J_s$ is compatible with this rotational invariance of T. (which is dense on the unit circle) and consider the punctured spheres $$X_m = \widehat{\mathbb{C}} \setminus \{e^{\pi si/2^n}, \ s = 0, 1, \dots, 2^{n+1} - 1\}, \quad m = 2^{n+1},$$ and their universal holomorphic covering maps $g_m: \mathbb{D} \to X_m$ normalized by $g_m(0) = 0, g'_m(0) > 0.$ The radial slits from the infinite point to all the points $e^{\pi si/2^n}$ form a canonical dissection $L_m$ of $X_m$ and define the simply connected surface $X'_m = X_m \setminus L_m$ . Any covering map $g_m$ determines a Fuchsian group $\Gamma_m$ of covering transformations uniformizing $X'_m$ , which act discontinuously in both disks $\mathbb{D}$ and $\mathbb{D}^*$ . Every such group $G_m$ has a canonical (open) fundamental polygon $P_m$ of $\Gamma_m$ in $\mathbb{D}$ corresponding to the dissection $L_m$ . It is a regular circular $2^{n+1}$ -gon centered at the origin of the disk and can be chosen to have a vertex at the point z=1. The restriction of $g_m$ to $P_m$ is univalent, and as $m \to \infty$ , these polygons entirely increase and exhaust the disk $\mathbb{D}$ . Similarly, we take in the complementary disk $\mathbb{D}$ the mirror polygons $P_m$ and the covering maps $g_m^(z) = 1/\overline{g_m(1/\overline{z})}$ which define the mirror surfaces $X_m$ . Now we approximate the maps $F^{\mu} \in \Sigma_{Q,\theta}$ by homeomorphisms $F^{\mu_m}$ having in $\mathbb{D} = \{|z| < 1\}$ the Beltrami coefficients $$\mu_m = [g_m]_* \mu := (\mu \circ g_m) \overline{g'_m} / g'_m, \quad n = 1, 2, \dots$$ Each $F^{\mu_m}$ is again k-quasiconformal (where $k = \|\mu\|_{\infty}$ ) and compatible with the group $\Gamma_m$ . As $m \to \infty$ , the coefficients $\mu_m$ are convergent to $\mu$ almost everywhere on $\mathbb{C}$ ; thus, the maps $F^{\mu_m}$ are convergent to $F^{\mu}$ uniformly in the spherical metric on $\widehat{\mathbb{C}}$ . Note also that $\mu_m$ depend holomorphically on $\mu$ as elements of $L_{\infty}$ ; hence, $F^{\mu_m}(0)$ is a holomorphic function of $t = F^{\mu}(0)$ . As a result, one obtains that the Beltrami coefficients $$\mu_{h,m} := [g_m]_* \mu_h$$ and the corresponding values $F^{\mu_{h,m}}(0)$ are holomorphic functions of the variable $t = F^{\mu}(0)$ . By Hartogs theorem, each function $\mathcal{J}_s(S_{F^{\mu_m}},t)$ with $t = F^{\mu_m}(0)$ is jointly holomorphic in $(S_{F^{\mu_m}},t) \in \mathcal{F}(\mathbf{T})$ . We now choose in $\mathbf{T}(0,m)\setminus\{\mathbf{0}\}$ represented as a subdomain of the space $\mathbf{B}(\Gamma_m)$ a countable dense subset $$E^{(m)} = \{\varphi_1, \varphi_2, \dots, \varphi_p, \dots\}.$$ For any of its point $\varphi_p$ , the corresponding extremal Teichüller disk $\mathbb{D}(\varphi_p)$ joining this point with the origin of $\mathbf{B}(\Gamma_m)$ does not meet other points from this set (this follows from the uniqueness of Teichüller extremal map). Recall also that each disk $\mathbb{D}(\varphi_p)$ is formed by the Schwarzians $S_{F^{\tau\mu_p;m}}$ with $|\tau| < 1$ and $$\mu_{p;m}(z) = |\psi_{p;m}(z)|/\psi_{p;m}(z)$$ with appropriate $\psi_{p;m} \in A_1(\mathbb{D}, \Gamma_m), \|\psi_{p;m}\|_1 = 1.$ The restrictions of the functionals $\mathcal{J}_s(S_{F^{\tau\mu_p;m}},t)$ to these disks are holomorphic functions of $(\tau,t)$ ; moreover, the above construction provides that all these restrictions are holomorphic in t in some common domain $D_m \subset \mathbb{D}_4$ containing the point t=0, provided that $|\tau| \leq k < 1$ . We use the maximal common holomorphy domain; it is located in a disk $\{|t| < r_0\}, \ r_0 < 4$ . Maximization over $\tau$ implies the logarithmically subharmonic functions $$U_{p,m}(t) = \sup_{|\tau| < 1} |\mathcal{J}_{1,2}(S_{F^{\tau\mu_{p,m}}}, t)| \quad (t = F^{\mu_{p,m}}(0), \quad p = 1, 2, \dots)$$ in the domain $D_m$ . We consider the upper envelope of this sequence $$u_m(t) = \sup_{p} \ U_{p;m}(t)$$ defined in some domain $D_m \subset \mathbb{D}_4$ containing the origin, and take its upper semicontinuous regularization $$u_m(t) = \limsup_{t' \to t} u_m(t'),$$ which does not increase max $|\mathcal{J}_{1,2}|$ ) (by abuse of notation, we shall denote the regularizations by the same letter as the original functions). Repeating this for all m, one obtains the sequences of monotone increasing functions $u_m(t)$ and of increasing domains $D_m$ exhausting a domain $D_\theta = \bigcup_m D_m$ such that each $u_m$ is subharmonic on $D_m$ , and the limit function of this sequence is equal to the function (22). It is defined and subharmonic on the domain $D_\theta$ . The lemma follows.

Lemma 11. [21] Let D be a bounded subdomain of $\mathbb{C}$ , G be a domain in a complex Banach space $X = \{\mathbf{x}\}$ and $\chi$ be a holomorphic map from G into the universal Teichmüller space $\mathbf{T} = \mathrm{Teich}(D)$ with the base point D (modeled as a bounded subdomain of $\mathbf{B}(D)$ ). Assume that $\chi(G)$ is a (pathwise connected) submanifold of finite or infinite dimension in $\mathbf{T}$ . Let w(z) be a holomorphic univalent solution of the Schwarz differential equation $$S_w(z) = \chi(\mathbf{x})$$ on D satisfying w(0) = 0, $w'(0) = e^{i\theta}$ with the fixed $\theta \in [-\pi, \pi]$ and $\mathbf{x} \in G$ (hence $w(z) = e^{i\theta}z + \sum_{n=0}^{\infty} a_n z^n$ ). Put $$|a_{2,\theta}^0| = \sup\{|a_2|: S_w \in \chi(G)\},$$ (23) and let $a_{2,\theta}^0 \neq 0$ and $w_0(z) = e^{i\theta}z + a_2^0z^2 + \dots$ be one of the maximizing functions. Then: (a) For every indicated function w(z), the image domain w(D) covers entirely the disk $D_{1/(2|a_{2,\theta}^0|)} = \{|w| < 1/(2|a_{2,\theta}^0|)\}.$ The radius value $1/(2|a_{2,\theta}^0|)$ is sharp for this collection of functions and fixed $\theta$ , and the circle $\{|w| = 1/(2|a_{2,\theta}^0|) \text{ contains points not belonging to } w(\mathbb{D}) \text{ if and only if } |a_2| = |a_{2,\theta}^0| \text{ (i.e., when } w \text{ is one of the maximizing functions).}$ (b) The inverted functions $$W(\zeta) = 1/w(1/\zeta) = e^{i\theta}\zeta - a_2^0 + b_1\zeta^{-1} + b_2\zeta^{-2} + \dots$$ with $\zeta \in D^{-1}$ map domain $D^{-1}$ onto a domain whose boundary is entirely contained in the disk $\{|W + a_{2,\theta}^0| \le |a_{2,\theta}^0|\}$ .

Lemma 12. The (image of) set $\Sigma_{af}$ is a three-dimensional subdomain in the space T. This important lemma was already applied in [18], [22]. Its proof is simple. For any function $F_{b_0^0,b_1^0,t^0}$ with $|b_0^0| < 2$ , $|b_1^0| < 1$ , $|t^0| < 1$ , there is a small neighborhood $U_0$ of the point $(b_0^0,b_1^0,t^0)$ in $\mathbb{C}^3$ such that all functions $F_{b_0,b_1,t}$ with $(b_0,b_1,t) \in U_0$ are univalent on $\mathbb{D}^*$ and admit quasiconformal extensions. The map $t \mapsto S_{ft}$ is holomorphic in B-norm, so the set $G_{af}$ is open in T. In addition, the line segment $[0, t^0]$ determines a curve in $G_{\text{af}}$ joining the point $F_{b_0^0, b_1^0, t^0}$ with the origin of T; hence $G_{\text{af}}$ is path-wise. Denote the image of $\Sigma_{\rm af}$ in Fib(T) by Fib( $G_{\rm af}$ ); it is a complex four-dimensional subdomanifold of Fib(T). The restriction of functionals $|J_s(S_{F^{\mu}},t)|$ to this set are plurisubharmonic on Fib( $\Sigma_{\rm af}$ ), and their maximization similar to Step 3 implies a maximal subharmonic function $u_{\rm af}(t)$ of type (22). Our goal now is to find $\max_{\Sigma_{af}} |\widetilde{J}_{1,2}(F)|$ . Since the corresponding quantity $$a_{2,J} := \max_{F_f \in \Sigma_{\text{af}}} |a_2(f)|$$ is positive, one can apply Lemmas 11 and 12. We select a dense subsequence $\{\theta_1, \theta_2, \dots\} \subset [-\pi, \pi]$ and define the corresponding functionals on the classes $S_{\theta_j}$ and $\Sigma_{\theta_j}$ replacing the original functionals $\widetilde{J}_s$ as follows. Having the functions $$f_a(z) = e^{i\theta_m}z + a_2z^2 + \dots$$ and the corresponding $$F_a(z) = 1/f_a(z) = e^{-i\theta_m}z + b_0 + b_1z^{-1} + \dots, \quad a = e^{i\theta_m}z$$ consider $z' = e^{i\theta_m}z$ as a new independent variable. Then $$f_a(z') = z' + e^{-2i\theta_m}(z')^2 + \dots, \quad F_a(z') = z' + b_0 e^{i\theta_m} + b_1 e^{i\theta_m}(z')^{-1} + \dots$$ belong to $S_Q$ and $\Sigma_Q$ (in terms of variable z'), and we set $$\widetilde{J}_{1,\theta_m}(F_a) = J_{1,\theta_m}(f_a) = \frac{e^{2inm\theta}a_n^2 - e^{i(2n-1)m\theta}a_{2n-1}}{(n-1)^2}, \quad \widetilde{J}_{2,\theta_m}(F_a) = J_{2,\theta_m}(f_a) = \frac{e^{inm\theta}a_n}{n}.$$ This preserves the weak rotational homogeneity. Note also that every $J_{s,\theta_m}(F_a)$ obeys Lemma 8, replacing (19) by the inequality $$|\widetilde{J}_s(F_a)| \ge \max_{b_0} |\widetilde{J}_s(F_{b_0,b_1;1}^{\theta})|,$$ where $$F_{b_0 b_0 t}^{\theta}(z) = e^{i\theta}z + b_0 t + b_1 t^2 z^{-1}$$ with $\theta = \arg a$ . (24) The above relations result in $$\max_{\Sigma_{\theta_j}} |\widetilde{J}_{s,\theta_j}| = \max_{\Sigma} |\widetilde{J}_s| = \max_{S} |J_s|, \tag{25}$$ and similarly for the corresponding collections $\Sigma_{\mathrm{af},\theta_i}$ of functions (24). Now consider the sequence of increasing products of the quotient spaces $$\mathcal{T}_m = \prod_{j=1}^m \widehat{\Sigma}_{\theta_j} / \sim = \prod_{j=1}^m \{ (S_{F_{\theta_j}}, F_{\theta_j}^{\mu_j}(0)) \} \simeq \mathbf{T}_1 \times \dots \times \mathbf{T}_1, \tag{26}$$ where the equivalence relation $\sim$ again means $\mathbf{T}_1$ -equivalence. The Beltrami coefficients $\mu_j \in \text{Belt}(\mathbb{D})_1$ are chosen here independently. For any $\mathbf{T}_1$ , presented in the right-hand side of (25), the corresponding values of $F_{\theta_j}^{\mu_j}(0)$ run over some domain $D_{\alpha_j} \subset \mathbb{C}$ , and the corresponding collection $\beta = (\beta_1, \dots, \beta_s)$ of the Bers isomorphisms $$\beta_j: \{(S_{W_{\theta_i}}, W_{\theta_i}^{\mu_j}(0))\} \to \mathcal{F}(\mathbf{T})$$ determines a holomorphic surjection of the space $\mathcal{T}_m$ onto the product of m spaces $\mathcal{F}(\mathbf{T})$ . Letting $$\mathbf{F}_{\theta^{\mu}}(0) := (F_{\theta_1}^{\mu_1}(0), \dots, F_{\theta_m}^{\mu_m}(0)), \quad \mathbf{S}_{\mathbf{F}_{\theta^{\mu}}} := (S_{F_{\theta_1}}, \dots, S_{F_{\theta_m}}),$$ consider the holomorphic maps (vector-functions) $$\mathbf{h}(\mathbf{S}_{\mathbf{F}_{\theta}}) = (h_1(S_{F_{\theta_1}}), \dots h_m(S_{F_{\theta_m}})) : \ \widehat{\Sigma}_{\mathrm{af},\theta} := \Sigma_{\mathrm{af},\theta_1} \times \dots \times \Sigma_{\mathrm{af},\theta_m} \to \mathbb{C}^m, \quad m = 1, 2, \dots,$$ with $$h_j(S_{F_{\theta_j}}) = \widetilde{Z}_{n,\theta_j}(F_a), \quad j = 1, \dots, m,$$ endowed with the polydisk norm $$\|\mathbf{h}\| = \max_{j} |h_{j}|$$ on $\mathbb{C}^m$ . Then by (25), $$\max_{\Sigma_{\text{af}}} \|\mathbf{h}\| = \max_{j} \max_{\Sigma_{\text{af}}} |h_{j}(S_{F_{\theta_{j}}})| = \max_{S} |J_{1,2}(f)|.$$ (27) The image of the set $\Sigma_{\rm af}$ under this embedding is the free product of m factors ${\rm Fib}_j(G_{\rm af})$ with dimension 4m. Note also that restriction of ${\bf h}$ to $\Sigma_{\rm af}$ is a polynomial map. We now apply the construction from Step 3 simultaneously to each component $h(S_{F_{\theta_j}}, t)$ on the corresponding space $\mathbf{T}_1$ in (26) and obtain similar to Lemma 9 that the function $$u_m(t) = \max(|h(S_{F_{\theta_1}}, t)|, \dots, |h(S_{F_{\theta_m}}, t)|)$$ is subharmonic in some domain $D_m$ containing the origin t = 0. This domain admits the rotational symmetry, hence it must be a disk $\mathbb{D}_{a_m}$ of some radius $a_m \leq 4$ . This symmetry follows from rotational symmetry of the set $\Sigma_{af}$ , inherited by its images in spaces $\mathbf{T}_1$ and $\mathcal{T}_m$ , and from Lemma 9 (applied, if needed, to functions $F \in \Sigma_Q$ and a prescribed set E in domain $F(\mathbb{D})$ ) varying F(0). We apply this lemma to functions $F \in \Sigma_Q$ and take the prescribed set E in domain $F(\mathbb{D})$ to vary F(0). Each function $u_m(t)$ is a circularly symmetric function on its disk $\mathbb{D}_{a_m}$ , and so is their upper envelope $$u_J(t) = \limsup_{m \to \infty} \widetilde{u}_m(t)$$ (on some disk $\mathbb{D}_a$ , a > 0). This envelope satisfies $$\max_{\mathbb{D}_a} u_J(t) = \max_{S} |J_{1,2}(f)|$$ and attains its maximal value at the boundary point t = a. Noting that the closure of $\Sigma_{\rm af}$ contains the functions $$F_{\theta}(z) = z - 2e^{i\theta} + e^{2i\theta}z^{-1}$$ inverting the Koebe functions $\kappa_{\theta}$ , one derives that the radius a must be equal 4, which means that the range domain of $F^{\mu}(0)$ for $F^{\mu} \in \Sigma_{\rm af}$ coincides with the disk $\mathbb{D}_4$ . This implies that the boundary points of this domain correspond only to functions f(z) with $|a_2| = 2$ , hence only to $\kappa_{\theta}(z)$ , and therefore, for all $f \in S$ with $F_f \in \Sigma_{\rm af}$ , we have the estimate $$|J_{1,2}(f)| \le |J_{1,2}(\kappa_{\theta})| = 1.$$ (28) Moreover, the equality in the left hand part occurs only when $f = \kappa_{\theta}$ . Step 5: Extremality of $\kappa_{\theta}$ on the whole class S. It remains to establish that the relations (28) are also valid for all $f \in S$ . Noting that the homotopy disk of Koebe's function $$\mathbb{D}(\kappa_{\theta}) = \{ \kappa_{\theta,t} = t^{-1} \kappa_{\theta}(tz) : |t| < 1 \}$$ is geodesic in universal Teichmüller space and that the functionals $J_1$ , $J_2$ generate on this disk a conformal metric equal to the hyperbolic metric of the unit disk (which follows from Step 4), one derives from asymptotic estimate (13) that this metric must be supporting at the origin for the maximal infinitesimal metric $\lambda_{J_{1,2}^0}(t)$ generated (via (19) applied separately to $J_1, J_2$ ) along the extremal map $f_0$ , since $\lambda_{J_{1,2}^0}(0) \leq \lambda_{\mathbb{D}}(0) = 1$ . Hence $f_0 = \kappa_{\theta}$ , which proves the extremality of $\kappa_{\theta}$ on the whole class S. One also has that $\kappa_{\theta}$ is unique extremal for $J_{1,2}$ on S, since $\lambda_{J_{1,2}^0}(t)$ majorates all conformal metrics determined by holomorphic maps $h: \mathbb{D} \to S$ on the corresponding disks $h(\mathbb{D}) \subset S$ , and accordingly, the corresponding integral distances generated by metrics $\lambda_{J_{1,2}^0}$ and $\lambda_h$ must satisfy for any $f \in S$ and any pair (t, z) the inequality $$|J_{1,2}^0(f(tz))| \le |J_{1,2}^0(\kappa_{\theta}(tz))| = (n-1)^2|t|^{2n-2}, \quad |t| \le 1,$$ where the equality (even on one pair (t, z)) arises only when $f(z) = \kappa_{\theta}(z)$ . This completes the proof of Theorem 1.