Hyperbolic metrics, homogeneous holomorphic functionals and Zalcman's conjecture

Theorem 1.1. Let J(f) be a homogeneous polynomial functional on S of the form (1.4) whose representation $\widetilde{J}(F_f)$ in the class $\Sigma$ does not contain free terms $c_db_0^d$ but contains nonzero terms with the coefficient $b_1$ of inversions $F_f$ . Then for all $f \in S$ , we have the sharp hound $$|J(f)| \le \max_{m} |J(\kappa_{m,\theta})|, \tag{1.9}$$ and this maximum is attained on some $\kappa_{m_0,\theta}$ ( $m_0 \geq 1$ ). If J has an extremal with $$b_1 = a_2^2 - a_3 \neq 0, (1.10)$$ then $|b_1| = 1$ and $$|J(f)| \le \max\{|J(\kappa_{\theta})|, |J(\kappa_{2,\theta})|\}. \tag{1.11}$$ The assumption (1.10) is equivalent to $$S_f(0) = -\lim_{z \to \infty} z^4 S_{F_f}(z) \neq 0.$$ The examples of some well-known functionals, for example, $J(f) = a_2^2 - \alpha a_3$ with $0 < \alpha < 1$ and $J(F_f) = b_m$ (m > 1), show that the assumptions on the initial coefficients $b_0$ and $b_1$ cannot be omitted. The Zalcman functional $$J_n(f) = a_n^2 - a_{2n-1}$$ is a special case of (1.4) with homogeneity degree 2n-2. For this functional, we obtain from Theorem 1.1 a complete result proving the Zalcman conjecture.

Theorem 1.2. For all $f \in S$ and any $n \geq 3$ , we have the sharp estimate (1.5), with equality only for $f = \kappa_{\theta}$ . As a consequence, one obtains also a new proof of the Bieberbach conjecture. Theorem 1.1 also provides other new distortion theorems concerning the higher coefficients. These results are presented in the last section. 1.4. It suffices to find the bound of J on functions admitting quasiconformal extensions across the unit circle and make the closure of this set in weak topology determined by locally uniform convergence on $\Delta$ . Such functions are naturally connected with the universal Teichmüller space $\mathbf{T}$ . The original functional J(f) is lifted to a holomorphic functional on a fiber space over $\mathbf{T}$ , but its growth is controlled by hyperbolic metrics on the base space $\mathbf{T}$ . Extremity of the Koebe function or of its root transforms is intrinsically connected with the features of these metrics on appropriate geodesic disks.

Proposition 2.1. [Kr4]. The infinitesimal Kobayashi metric $\mathcal{K}_{\mathbf{T}}(\psi, v)$ on the tangent bundle $\mathcal{T}(\mathbf{T})$ of the universal Teichmüller space $\mathbf{T}$ is logarithmically plurisubharmonic in $\psi \in \mathbf{T}$ , equals the canonical Finsler structure $F_{\mathbf{T}}(\psi, v)$ on $\mathcal{T}(\mathbf{T})$ generating the Teichmüller metric of $\mathbf{T}$ and has constant holomorphic sectional curvature -4. It implies that the Teichmüller distance $\tau_{\mathbf{T}}(\varphi, \psi)$ is logarithmically plurisubharmonic in each of its variables and hence the pluricomplex Green function of the space $\mathbf{T}$ equals $$g_{\mathbf{T}}(\varphi, \psi) = \log \tanh \tau_{\mathbf{T}}(\varphi, \psi) = \log k(\varphi, \psi),$$ (2.5) where $k(\varphi, \psi)$ denotes the extremal dilatation of quasiconformal maps determining the Teichmüller distance between the points $\varphi$ and $\psi$ in $\mathbf{T}$ . Recall that the pluricomplex Green function $g_D(x,y)$ of a domain D in a complex Banach manifold X with pole y is defined by $g_D(x,y) = \sup u_y(x)$ $(x,y \in D)$ and followed by upper regularization $v^*(x) = \limsup_{x' \to x} v(x')$ , taking the supremum over all plurisubharmonic functions $u_y(x): D \to [-\infty, 0)$ such that $$u_y(x) = \log ||x - y||_X + O(1)$$ in a neighborhood of the pole y. Here $\|\cdot\|_X$ denotes the norm on the space modeling X, and the remainder term O(1) is bounded from above. The Green function $g_D(x,y)$ is a maximal plurisubharmonic function on $D \setminus \{y\}$ (unless it is identically $-\infty$ ). (c) The assumption $F^{\mu}(0) = 0$ for quasiconformal extensions of a function $F \in \Sigma^0$ made above ensures nonvanishing F in $\Delta^*$ (and $f(z) = 1/F(1/z) \in S$ ). In addition, completely normalized maps $F^{\mu}(z)$ are holomorphic functions of their Beltrami coefficients $\mu \in \mathbf{Belt}(\Delta)_1$ as well as of their Schwarzians. The same holds for the coefficients $b_m$ of $F^{\mu}$ . The Schwarzian equations $S_w = \varphi$ for $F \in \Sigma$ and for its inversion $f \in S$ and the Beltrami equation for quasiconformal extensions of these functions determine their solutions up to linear transformations, which for F are the translations $w \mapsto w + b_0$ and for f have the form $$w \mapsto w/(1 - \alpha w) = w + \alpha w^2 + \cdots$$ with $\alpha$ determined by the coefficients $a_2$ . The admissible values of $b_0 = -a_2$ , which are the free terms of corresponding $F_1 \in \Sigma$ having the same Schwarzian, are only those which range over the closed domain $\widehat{\mathbb{C}} \setminus F(\Delta^*)$ . Since the original normalization of the functions from $\Sigma$ and S includes only two conditions, the initial functional $J_n$ must be considered on the fiber space $\mathcal{F}(\mathbf{T})$ over $\mathbf{T}$ , which is modeled as a bounded domain in the space $\mathbf{B} \times \Delta(0,2)$ , whose points are the pairs $(\varphi, a)$ where $\varphi$ are the Schwarzians of $F \in \Sigma^0$ with F(0) = 0 and a are equal to the second coefficients $a_2$ of their inversions in S. The defining projection of this space $$\pi_{\mathcal{F}}: (S_{F^{\widetilde{\mu}}}, a_2^{\mu}) \to S_{F^{\widetilde{\mu}}}$$ is a holomorphic split submersion. The fibers $\pi_{\mathcal{F}}^{-1}(\varphi)$ over the base points $\varphi = S_F$ coincide with the complementary domains $\widehat{\mathbb{C}} \setminus \overline{F(\Delta^*)}$ , giving the admissible values of $a_2$ . Note also that the space $\mathcal{F}(\mathbf{T})$ is biholomorphically isomorphic to the Bers fiber space over $\mathbf{T}$ (both fiber spaces have the same base and differ only by an additional normalization of maps $F \in \Sigma$ with a given Schwarzian $\varphi = S_F$ ) and thus is isomorphic to the Teichmüller space of the punctured disk $\Delta \setminus \{0\}$ (cf. [Be2]). 2.2. The Grunsky operator. The complex geometry of the universal Teichmüller space is closely connected with the Grunsky inequalities technique which arose from investigating the univalence problem in [Gr]. Any function $F \in \Sigma$ (and similarly any $f \in S$ ) determines its Grunsky operator (matrix) $\mathcal{G}_F = (\alpha_{mn}(F))$ , where the Grunsky coefficients $\alpha_{mn}$ are determined by the expansion $$\log \frac{F(z) - F(\zeta)}{z - \zeta} = -\sum_{m,n=1}^{\infty} \alpha_{mn} z^{-m} \zeta^{-n}, \quad (z,\zeta) \in (\Delta^*)^2,$$ with the principal branch of logarithmic function, satisfy the inequalities $$\left|\sum_{m,n=1}^{\infty} \sqrt{mn} \alpha_{mn} x_m x_n\right| \le k. \tag{2.6}$$ Here $\mathbf{x} = (x_n)$ runs over the unit sphere $S(l^2)$ of the Hilbert space $l^2$ with norm $\|\mathbf{x}\| = \left(\sum_{i=1}^{\infty} |x_n|^2\right)^{1/2}$ , and $k = k(F) \le 1$ is the Teichmüller norm of F (cf. [Gr], [Ku1]). The quantity $$\varkappa(F) = \sup\left\{ \left| \sum_{m,n=1}^{\infty} \sqrt{mn} \ \alpha_{mn} x_m x_n \right| : \mathbf{x} = (x_n) \in S(l^2) \right\} \le 1$$ is called the Grunsky norm of F. It equals the norm of $\mathcal{G}_F$ regarded as a linear operator $l^2 \to l^2$ . The functions with $\varkappa(F) = k(F)$ play a crucial role in applications of Grunsky inequalities; however, the set of $S_F$ , on which $\varkappa(F) < k(F)$ , is open and dense in T. One of the underlying facts in applications is the following result.

Proposition 2.2. The equality $\varkappa(F) = k(F)$ for $f \in \Sigma^0$ holds if and only if the function F is the restriction to $\Delta^*$ of a quasiconformal self-map $w^{\mu_0}$ of $\widehat{\mathbb{C}}$ with Beltrami coefficient $\mu_0$ satisfying the condition $\sup |\langle \mu_0, \psi \rangle_{\Delta}| = \|\mu_0\|_{\infty}$ , where the supremum is taken over holomorphic functions $\psi \in A_1^2(\Delta)$ with $\|\varphi\|_{A_1(\Delta)} = 1$ , where $$A_1^2 = \{ \psi \in A_1(\Delta) : \psi = \omega^2 \text{ with } \omega \text{ holomorphic on } \Delta.$$ In addition, if the equivalence class [F] contains a frame map, i.e., is a Strebel point (see Section 2.4), then the restriction of $\mu_0$ onto the disk $\Delta$ must be of the form $$\mu_0(z) = k|\psi_0(z)|/\psi_0(z) \quad \text{with } \psi_0 \in A_1^2.$$ (2.7) The proof of this proposition is given in [Kr2], [Kr7]. It relies on the fact that the Grunsky coefficients $\alpha_{mn}(S_F)$ generate the holomorphic functions $$h_{\mathbf{x}}(\varphi) = \sum_{m,n=1}^{\infty} \sqrt{mn} \, \alpha_{mn}(\varphi) x_m x_n, \qquad (2.8)$$ where $\varphi = S_f$ and $\mathbf{x} = (x_n)$ are the points of the shere $S(l^2)$ , mapping the universal Teichmüller space $\mathbf{T}$ into the unit disk $\Delta$ . The restrictions of these functions to the disk $\{\phi_{\mathbf{T}}(s\mu_0)\}$ determine the Carathéodory distance between the points $S_{f^{s\mu_0}}$ and the origin, which by (2.6) equals the Teichmüller distance. In a special case, when the curve $F(S^1)$ is analytic, the equality (2.7) was obtained by a different method in [Ku2]. In particular, the maps $$F_{m,t}(z) = \frac{1}{\kappa_{m,t}(1/z)} = z \left(1 - \frac{t}{z^{m+1}}\right)^{2/(m+1)} = z - \frac{2t}{m+1} \frac{1}{z^m} + \dots, \quad |t| \le 1,$$ (2.9) whose extremal extensions to $\mathbb{C}$ have Beltrami coefficients $$\mu_{F_{m,t}}(z) = t|z|^{m-1}/z^{m-1}$$ for $|z| < 1$ , satisfy $\varkappa(F_{m,t}) = k(F_{m,t})$ for odd $m \ge 1$ and $\varkappa(F_{m,t}) < k(F_{m,t})$ for even $m \ge 0$ . Note that holomorphy of the functions (2.8) is a consequence of the fact that the Grunsky coefficients $\alpha_{mn}$ are polynomials of the initial coefficients $b_1, \ldots, b_{m+n-1}$ of F combined with the well-known inequality (cf. [Po, p. 61]): for any $1 \le p \le M$ , $1 \le q \le N$ , $$\left| \sum_{m=p}^{M} \sum_{n=q}^{N} \sqrt{mn} \, \alpha_{mn} x_m x_n \right|^2 \le \sum_{m=p}^{M} |x_m|^2 \sum_{n=q}^{N} |x_n|^2.$$ We mention also that both Teichmüller and Grunsky norms are continuous logarithmically plurisubharmonic functions on T (see [Kr6], [Sh]) and that, by a theorem of Pommerenke and Zhuravlev, any $F \in \Sigma$ with $\varkappa(F) \leq k < 1$ has $k_1$ -quasiconformal extensions to $\widehat{\mathbb{C}}$ with $k_1 = k_1(k) \geq k$ (see [Po]; [KK1, pp. 82-84], [Zh]). 2.3. A holomorphic homotopy of univalent function. Similarly to the functions in S, one can define for each $F \in \Sigma$ with expansion (1.1) the complex homotopy $$F_t(z) = tF\left(\frac{z}{t}\right) = z + b_0 t + b_1 t^2 z^{-1} + b_2 t^3 z^{-2} + \dots : \Delta^* \times \Delta \to \widehat{\mathbb{C}}$$ (2.10) so that $F_0(z) \equiv z$ . This implies $$S_{F_t}(z) = t^{-2} S_F(t^{-1}z),$$ and moreover, this point-wise map determines a holomorphic map $$h_F(t) = S_{F_t}(\cdot): \Delta \to \mathbf{B}$$ (2.11) (see, e.g. [Kr3]). The corresponding homotopy disk $$\Delta(S_F) = h_F(\Delta) \subset \mathbf{T} \tag{2.12}$$ is holomorphic at noncritical points of maps (2.11). These disks foliate the space T (and the set $\Sigma$ ). The dilatations of the homotopy maps are estimated by

Proposition 2.3. [Kr3] (a) Each homotopy map $F_t$ of $F \in \Sigma$ admits k-quasiconformal extension to the complex sphere $\widehat{\mathbb{C}}$ with $k \leq |t|^2$ . The bound $k(F_t) \leq |t|^2$ is sharp and occurs only for the maps $$F_{b_0,b_1}(z) = z + b_0 + b_1 z^{-1}, \quad |b_1| = 1,$$ whose homotopy maps $$F_{b_0,b_1t^2}(z) = z + b_0t + b_1t^2z^{-1} (2.13)$$ have the affine extensions $\widehat{F}_{b_0,b_1t^2}(z) = z + b_0t + b_1t^2\overline{z}$ onto $\Delta$ . (b) If $F(z) = z + b_0 + b_m z^{-m} + b_{m+1} z^{-(m+1)} + \dots$ ( $b_m \neq 0$ ) for some integer m > 1, then the minimal dilatation of extensions of $F_t$ is estimated by $k(F_t) \leq |t|^{m+1}$ ; this bound also is sharp. In the second case. $$h_F(0) = h'_F(0) = \dots = h_F^{(m)}(0) = \mathbf{0}, \ h_F^{(m+1)}(0) \neq \mathbf{0},$$ and due to [KK2], $$k(F_t) = \frac{m+1}{2} |b_m| |t|^{m+1} + O(t^{m+2}), \quad t \to 0.$$ (2.14) This bound is sharp. The simplest holomorphic disks in $\Sigma^0$ are the images of Teichmüller extremal disks $$\Delta(\psi) = \{\phi_{\mathbf{T}}(t\mu_0) : t \in \Delta\} \subset \mathbf{T}$$ formed by functions F whose extremal extensions onto $\Delta$ (with minimal dilatation) have Beltrami coefficients $t\mu_0$ with $$\mu_0(z) = |\psi(z)|/\psi(z),$$ (2.15) where $\psi$ is a holomorphic function from $L_1(\Delta)$ . Such an extension is unique (up to a constant factor of $\psi$ and normalization of F). The Teichmüller disks foliate dense subsets in $\mathbf{T}$ and $\Sigma^0$ . Note also that the homotopy function $F_t$ has such extremal extension for each $t \in \Delta$ (cf. [GL], [St]) and that for any function (2.9) its homotopy disk $\Delta(S_{F_m})$ is of Teichmüller type.

Lemma 3.1. Let X be a complex Banach manifold with complete Kobayashi and Carathéodory metrics, and let $h: \Delta \to X$ be a complex geodesic. If the restriction of a holomorphic map $X \to \Delta$ to the disk $D = h(\Delta)$ has at $x_0 = h(0)$ zero of order $p \ge 1$ , then for all $t \in \Delta$ , $$|j \circ h(t)| \le \tanh d_X(h(t^p), \mathbf{0}).$$ (3.6) The equality (even for one $t \neq 0$ ) only occurs for defining functions of the Carathéodory distance between the points $x_0$ and $x \in D$ , then $$d_{\Delta}(j \circ h(t^p)), j \circ h(0)) = c_X(h_0(t^p), h(0))$$ for all $t \in \Delta$ . Proof. First recall the Schwarz lemma for subharmonic functions (we only need its simplest version).

Lemma 3.2. Let a function $u(t): \Delta \to [0,1)$ be logarithmically subharmonic in the disk $\Delta$ and such that the ratio $u(t)/|t|^m$ is bounded in a neighborhood of the origin for some $m \ge 1$ . Then $$u(t) < |t|^m \quad for \ all \ \ t \in \Delta$$ (3.7) and $$\limsup_{|t| \to 0} \frac{u(t)}{|t|^m} \le 1. \tag{3.8}$$ Equality in (3.7), even for one $t_0 \neq 0$ , or in (3.8), can only hold for the function $u(t) = |t|^m$ . The assumption on metrics $d_X$ and $c_X$ yields that the pluricomplex Green function $g_X(x, y)$ of X with a pole at y cannot be equal identically $-\infty$ and that for any pair of points $x, y \in X$ with equal Kobayashi and Carathéodory distances, $$g_X(x,y) = \log \tanh d_X(x,y) = \log \tanh c_X(x,y); \tag{3.9}$$ in addition, $\lim g_X(x,y) = 0$ as x tends to infinity (boundary of X), for any fixed $y \in X$ . From (3.9) and Lemma 3.2, $$g_X(h(t^p), \mathbf{0}) = \log |t|^p$$ for all $|t| < 1$ . The function $j \circ h(t)$ is logarithmically subharmonic on $\Delta$ and has zero of order p at the origin, hence by the same Schwarz's lemma, $$\log|j \circ h(t)| \le \log|t|^p,$$ which implies the estimate (3.6), completing the proof of Lemma 3.1. We proceed to the proof of Theorem 1.1 and denote by s the canonical complex parameter on the Teichmüller disks in in $\mathcal{F}(\mathbf{T})$ generated by admissible (that is, nonvanishing on $\Delta^*$ ) functions $$F_{b_0,s}(z) = z + b_0 + sz^{-1},$$ whose extremal extensions onto $\overline{\Delta}$ are the affine maps $z \mapsto z + b_0 + s\overline{z}$ . These disks cover $\Delta(S_{F_0,s}) \subset \mathbf{T}$ . If such $F_{b_0,s}$ is admissible only for $|s| < s_0 < 1$ , one can reparametrize it using the parameter $\sigma = s/s_0$ which runs over the unit disk. For each $b_0$ , $$d_{\mathbf{T}_1}(\mathbf{0}, (S_{F_{b_0,s}}, b_0)) = d_{\mathbf{T}}(\mathbf{0}, S_{F_{0,s}}); \tag{3.10}$$ in addition, every map $$\sigma \mapsto (S_{F_{b_0,\sigma}}, b_0 \sigma), \quad \sigma \in \mathbf{T},$$ determines a complex geodesics in the space $\mathcal{F}(\mathbf{T})$ . Since by (2.14) the parameters s and t are related near the origin by $$s = b_1 t^2 + O(t^3)$$ as $t \to 0$ and $b_1 \neq 0$ , it follows from (3.5) that the restrictions of the functions (3.4) to any disk $\Delta(S_{Fb_0,s})$ have zero of order d/2 at the origin. Lemma 3.1 implies the bound $$|g_m(S_{F_{b_0,s}})| \le [\tanh d_{\mathbf{T}}(S_{F_{b_0,s}}, \mathbf{0})]^{d/2} = |s|^{d/2},$$ or equivalently, $$|g_m(S_{F_{0,b_1t^2}})| \le |b_1|^{d/2}|t|^d + O(|t|^{d+1}), \quad t \to 0.$$ (3.11) Note that by (3.3) the remainder in (3.11) does not depend on m. Therefore, $$\mathcal{J}(S_{F_{0,b_1t^2}}) = \sup_{m} |g_m(S_{F_{0,b_1t^2}})| \le |b_1|^{d/2} |t|^d + O(|t|^{d+1}). \tag{3.12}$$ The relations (3.5) and (3.12) imply the equality $$\mathcal{J}(S_F) \le |b_1|^{d/2} = \left(\frac{|S_f(0)|}{6}\right)^{d/2} \le 1. \tag{3.13}$$ Now observe that if the original functional J(f) has an extremal $f_0(z) = z + \sum_{n=1}^{\infty} a_n^0 z^n$ , for which the assumption (1.10) is fulfilled, then the value of the left-hand side in (3.21) on this function must be equal to 1, because in this case, $$\frac{|\widetilde{J}(S_{F_0}, a_2^0)|}{M(J)} = \mathcal{J}(S_{F_0}) = 1,$$ thus $$\frac{1}{6}|S_{f_0}(0)| = |(a_2^0)^2 - a_3^0| = 1.$$ As was mentioned, such equalities can only occur when $f_0$ either is the Koebe function $\kappa_{\theta}$ or it coincides with the odd function $\kappa_{2,\theta}$ defined by (1.6). In addition, the extremality of $f_0$ implies $$|J(f_0)| = M(J) = \max\{|J(\kappa_{\theta})|, |J(\kappa_{2,\theta})|\}.$$ (b) The functions $f \in S$ with $S_f(0) = 0$ omitted above can be approximated (in B-norm) by f with $S_f(0) \neq 0$ by applying special quasiconformal deformations of the plane given by the following lemma from [Kr1, Ch. 4]. This lemma softens the strongest rigidity of conformal maps.

Lemma 3.3. In a finitely connected domain $D \subset \widehat{\mathbb{C}}$ , let there be selected a set E of positive two-dimensional Lebesgue measure and the distinct finite points $z_1, \ldots, z_n$ with assigned nonnegative integers $\alpha_1, \ldots, \alpha_n$ , respectively, so that $\alpha_j = 0$ for $z_j \in E$ . Then, for sufficiently small $\varepsilon > 0$ and $\varepsilon \in (0, \varepsilon_0)$ , for any given system of numbers $\{w_{sj}\}$ , $s = 0, 1, \ldots, \alpha_j$ , $j = 1, \ldots, n$ , such that $w_{0j} \in D$ , $$|w_{0j} - z_j| \le \varepsilon$$ , $|w_{1j} - 1| \le \varepsilon$ , $|w_{sj}| \le \varepsilon$ $(s = 2, \dots, \alpha_j, j = 1, \dots, n)$ , there exists a quasiconformal automorphism $h_{\varepsilon}$ of the domain D, which is conformal on the set $D \setminus E$ and satisfies $h_{\varepsilon}^{(s)}(z_j) = w_{sj}$ for all $s = 0, 1, ..., \alpha_j$ and j = 1, ..., n, with dilatation $\|\mu_{h_{\varepsilon}}\|_{\infty} \leq M\varepsilon$ . The constants $\varepsilon_0$ and M depend only on D, E and the vectors $(z_1, ..., z_n)$ , $(\alpha_1, ..., \alpha_n)$ . If the boundary $\Gamma$ of domain D is Jordan or belongs to the class $C^{l,\alpha}$ , where $0 < \alpha < 1$ and $l \ge 1$ , one can take $z_i \in \Gamma$ with $\alpha_i = 0$ or $\alpha_i \le l$ , respectively. Now, let $f \in S^0$ have coefficients $a_2$ and $a_3$ related by $a_3 = a_2^2$ , i.e., $b_1(f) := b_1(F_f) = 0$ . Since $f(\Delta^*)$ is a domain, one can take there a set E of positive measure and construct by Lemma 3.5 for a sequence $\varepsilon_n \to 0$ such variations $h_n = h_{\varepsilon_n}$ of f that for each n, $$b_1(h_n \circ f) = b_1(f) + O(\varepsilon_n) \neq 0, \quad |J(h_n \circ f)| = |J(f)| + O(\varepsilon_n) > |J(f)|.$$ Since, by the previous step. $$|J(h_n \circ f)| \le \max\{|J(\kappa_\theta)|, |J(\kappa_{2,\theta})|\},$$ the same estimate will hold also for f. (c) Finally, consider the case when J has no extremals $f_0$ satisfying (1.10), and hence any extremal inversion $F_{f_0}$ is of the form $$F(z) = z + b_0 + b_m z^{-m} + b_{m+1} z^{-(m+1)} + \dots \qquad (b_m \neq 0; |z| > 1)$$ (3.14) with m > 1. If m + 1 does not divide d = d(J), we consider the functional $J^{m+1}$ , which is d(m+1)-homogeneous; otherwise one can use J. Let us show that one can apply to $J^{m+1}$ the above arguments, replacing $F_{0,b_1t^2}$ by the corresponding function $F_{m,t}$ represented by (2.9). Its Schwarzian relates to $S_{F_t}$ by $$S_{F_t} = S_{F_{m,t}} + O(t^{m+1}), \quad t \to 0.$$ If m is odd, one immediately derives from Proposition 2.2 (applied to $\mu_{F_{m,t}}(z) = t|z|^{m-1}/z^{m-1}$ ) that $$c_{\mathbf{T}}(\mathbf{0}, S_{F_{m,t}}) = d_{\mathbf{T}}(\mathbf{0}, S_{F_{m,t}}) = \tau_{\mathbf{T}}(\mathbf{0}, S_{F_{m,t}}) = \tanh^{-1}\left(\frac{m+1}{2}|b_m||t|^{m+1}\right).$$ (3.15) If m is even, we consider the map $$F_2(z) = F(z^2)^{1/2} = z + \frac{b_0}{2} \frac{1}{z} + \frac{b_m}{2} \frac{1}{z^{2m-1}} + \dots$$ which is well defined, since F(0) = 0, and represents an odd function symmetric with respect to the origin. Denote the Taylor coefficients of $F_2$ by $b_j^{(2)}$ and let $\alpha_{mn}^{(2)}(F) = \alpha_{mn}(F_2)$ . Squaring $$\mathcal{R}_2: F(z) \mapsto F(z^2)^{1/2}$$ transforms the quadratic differentials $\psi = \psi(z)dz^2$ in $\Delta$ into $\mathcal{R}_2^\psi = \psi_r(z^2)4z^2dz^2$ , which have zero of even order at the origin. Thus one can apply Proposition 2.2 to $F_2$ , using instead of (2.8) the functions $$h_{2,\mathbf{x}}(\varphi) = \sum_{p,q=1}^{\infty} \sqrt{pq} \ \alpha_{pq}^{(2)}(\varphi) x_p x_p : \ \mathbf{T} \to \Delta.$$ (3.16) Indeed, all maps f and $F_f$ are completely normalized and their Beltrami coefficients and Schwarzians are related by $$\mu_F(z) = \mu_f(1/z)z^2/\overline{z}^2, \quad S_F(z) = -S_f(1/z)z^{-2}.$$ Applying the Cauchy formula for derivatives of holomorphic functions, one derives that the Taylor coefficients $a_n$ , $n \geq 2$ , and $b_j$ , $j \geq 0$ , depend holomorphically on Beltrami coefficients $\mu_F \in \mathbf{Belt}(\Delta)_1$ and on Schwarzians $S_F \in \mathbf{T}$ . On the other hand, each of the coefficients $b_j^{(2)}$ and $\alpha_{pq}^{(2)}$ of $F_2$ is represented as a polynomial of a finite number of initial coefficients $b_0, b_1, \ldots, b_s$ of the original function F (noting that the free term $b_0$ is uniquely determined by assumption F(0) = 0). Thus the maps (3.16) also depend holomorphically on $\mu_F$ and $S_F$ . Finally, the transform $\mathcal{R}_2$ preserves quasiconformal dilatations. These properties imply that $S_{F_{m,t}}$ satisfies all equalities in (3.15) and hence ranges in a geodesic disk in T. Combining with (2.14), one now obtains instead of (3.13) the bound $$\mathcal{J}(S_F)^{m+1} \le \left(\frac{m+1}{2}|b_m|\right)^d \le 1,$$ or $$\mathcal{J}(S_F) \le \left(\frac{m+1}{2}|b_m|\right)^{d/(m+1)} \le 1. \tag{3.17}$$ To examine the case of equality in (3.17), observe that since $\mathcal{J}(S_{F_{f_0}}) = 1$ for any extremal $f_0$ , (3.17) yields (denoting the coefficients of $F_{f_0}$ by $b_j^0$ ) $$\frac{m+1}{2}|b_m^0| = 1.$$ The functions (3.13) with m > 1 satisfy $b_1 = \cdots = b_{m-1} = 0$ , which allows us to estimate $b_m$ using the well-known inequalities of Golusin and Jenkins (see [Go, Ch. XI], [Je]). These inequalities hold for a more general univalent $F(z) = z + \sum_{0}^{\infty} b_n z^{-n}$ , |z| > 1 (but not for all $F \in \Sigma$ ), and in our special case imply the bound $$|b_m| \le 2/(m+1)$$ . Moreover, the equality here only occurs for the function (2.9) with |t| = 1 (and its admissible translations $F_{m,t}(z) + c$ ), which also implies $M(J) = |J(\kappa_{m,\theta})|$ . We have established that any extremal function $f_0$ maximizing |J(f)| must be of the form (1.8). The theorem is proved. Remark. Lemma 3.1 is interesting for its own sake and has other applications. Thus we also present a different proof which does not involve Green's function and Lemma 3.2. By assumption, $$g \circ h(t) = c_p t^p + c_{n+1} t^{p+1} + \dots \ (c_p \neq 0).$$ Let $\{g_m\}$ be a maximizing sequence for the Carathéodory distance $c_X(x_0, x)$ with $g_m(x_0) = 0$ , where $x \in D = h(\Delta)$ is distinct from $x_0$ , i.e., $$c_X(x_0, x) = \lim_{m \to \infty} d_{\Delta}(0, g_m(x)).$$ The restrictions of $g_m$ to D are convergent locally uniformly on D to a holomorphic function $g_0$ on this disk with $g_0(x_0) = 0$ so that $d_{\Delta}(0, g_0(x)) = c_X(x_0, x)$ , $x \in D$ . Since both metrics $c_X$ and $d_X$ are hyperbolic isometries between $\Delta$ and D, the map $g_0 \circ h$ is the identity on $\Delta$ ; hence, $g_0^p \circ h(t) = t^p$ for all $t \in \Delta$ , and $$|j \circ h(t)| \le |g_0^p \circ h(t)|. \quad t \in \Delta.$$ Together with the equality $g_0 \circ h(t^p) = t^p$ , $t \in \Delta$ , this implies (3.6). The case of equality in (3.6) follows from Schwarz's lemma for holomorphic functions with zero of a prescribed order at the origin.

Proposition 4.1. [Kr5] For all $f(z) = z + \sum_{n=0}^{\infty} a_n z^n \in S_{\infty}(k)$ and all $k \leq 1/(n^2 + 1)$ , we have the sharp bound $$|a_n| \le \frac{2k}{n-1},\tag{4.2}$$ with equality only for the functions $$f_{n-1,t}(z) = f_{1,t}(z^{n-1})^{1/(n-1)} = z + \frac{2kt}{n-1}z^n + \dots, \quad n = 3, 4, \dots$$ (4.3) Note that every function (4.3) admits a quasiconformal extension onto $\Delta^*$ with Beltrami coefficient $\mu_n(z) = t|z|^{n+1}/z^{n+1}$ and $\widehat{f}_{n-1,t}(\infty) = \infty$ . Accordingly, $F_{n-1,t}(z) = 1/f_{n-1,t}(1/z) \in \Sigma^0$ admits a quasiconformal extension onto the unit disk with $F_{n-1,t}(0) = 0$ and $\mu_{F_{n-1,t}}(z) = t|z|^{n-1}/z^{n-1}$ for |z| < 1. Another essential point is that for any function $$F_{n-1}(z) := F_{n-1,1}(z) = 1/\kappa (1/z^{n-1})^{1/(n-1)},$$ its homotopy disk $\Delta(S_{F_{n-1}})$ in T is of Teichmüller type. Together with estimate (4.2), this implies that for any m > 2 and small r > 0, $$|J_n(\kappa_{m,r})| < r(n-1)^2;$$ thus $$|J_n(\kappa_{m,\theta})| < (n-1)^2 = J_n(\kappa),$$ completing the proof of Theorem 1.2. Remark. The above arguments work well also in the case of functionals obtained by suitable perturbation of $J_n(f)$ . For example, one can take $$J(f) = a_n^2 - a_{2n-1} + P(a_3, \dots, a_{2n-2}),$$ where P is a homogeneous polynomial of degree 2n-2, $$P(a_3,\ldots, a_{2n-2}) = \sum_{|k|=2n-2} c_{k_3,\ldots,k_n} a_3^{k_2} \ldots a_n^{k_{2n-2}},$$ and $|k| := k_3 + \ldots + k_{2n-2}$ , $a_j = a_j(f)$ , assuming that this polynomial has nonnegative coefficients and satisfies $$\max_{S} |P(a_3, \dots, a_{2n-2})| < \frac{(n-1)^2}{2}.$$ For any such functional, only the Koebe function is extremal.

Theorem 5.1. For all $f \in S$ and integers n > 3 and $p \ge 1$ , $$|a_n^p - a_2^{p(n-1)}| \le 2^{p(n-1)} - n^p.$$ This bound is sharp, and the equality only occurs for the Koebe function $\kappa_{\theta}$ . Proof. Since $b_0 = -a_2$ , the relation (1.3) yields $$I_n(f) := a_n - a_2^{n-1} = (n-2)(-1)^{n-1}b_1b_0^{n-3} + \text{lower terms with respect to } b_0.$$ This functional satisfies the assumptions of Theorem 1.1. The same arguments as in the proof of Theorem 1.2 imply $$|I_n(\kappa_{m,\theta})| < |I_n(\kappa_{\theta})|$$ for all $m \ge 2$ , completing the proof. In the same way, one obtains

Theorem 5.2. For all $f \in S$ and integers n > 2 and $p \ge 1$ , $$|a_{n+1}^p - a_2^p a_n^p| \le 2^p n^p - (n+1)^p$$ with equality only for $f = \kappa_{\theta}$ .