A short geometric proof of the Zalcman and Bieberbach conjectures

Theorem 2.1. Let J(f) be a homogeneous polynomial functional on S of the form (2.4) 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 bound $$|J(f)| \le \max_{m} |J(\kappa_{m,\theta})|, \tag{2.2}$$ and this maximum is attained on some $\kappa_{m_0,\theta}$ $(m_0 \ge 1)$ . If J has an extremal with $$b_1 = a_2^2 - a_3 \neq 0, \tag{2.3}$$ then $|b_1| = 1$ and $$|J(f)| \le \max\{|J(\kappa_{\theta})|, |J(\kappa_{2,\theta})|\}. \tag{2.4}$$ The assumption (2.3) is equivalent to $$S_f(0) = -\lim_{z \to \infty} z^4 S_{F_f}(z) \neq 0,$$ where $S_f$ denotes the Schwarzian derivative of f in $\Delta$ defined by $$S_f = (f''/f')' - (f''/f')^2/2.$$ 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.

Theorem 2.2. 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 2.1 also provides other new distortion theorems concerning the higher coefficients. These results are presented in Section 6. In the last section, we show how the proof of Theorem 2.1 yields asymptotic estimating the growth rate of generic homogeneous functionals on an individual function with quasiconformal extension. 2.3. Connection with geometry of Teichmüller spaces. Our approach to these problems is geometric. Its origins go back to [Kr7] where the proof of Zalcmann's conjecture for the initial coefficients was given. It suffices to find the bound of J on functions f admitting quasiconformal extensions across the unit circle and close this set in weak topology determined by locally uniform convergence on $\Delta$ . Denote the subset of such f by $S^0$ and the set of corresponding $F_f \in \Sigma$ by $\Sigma^0$ . Such functions are naturally connected with the universal Teichmüller space $\mathbf{T} = \mathbf{T}(\Delta)$ and the Teichmüller space $\mathbf{T}_1 = \mathbf{T}(\Delta \setminus \{0\})$ of the punctured disk. Accordingly, the original functional J(f) is lifted to a holomorphic functional on $\mathbf{T}_1$ , and its sharp upper bound is obtained using deep geometric features of this space. Application of metrics of negative generalized curvature in the lines of [Kr7] allows us to estimate the functional from below giving the same asymptotic bound. In fact, we establish that every homogeneous holomorphic functional on S satisfying the assumptions of Theorem 2.1 determines a complex geodesic in the space $\mathbf{T}_1$ generated by some $\kappa_{m,\theta}$ .

Proposition 3.1. [Kr9] If the restriction of a holomorphic map $h: \widetilde{\mathbf{T}} \to \Delta$ onto a geodesic disk $\Delta(\mu_0) = \{\phi_{\widetilde{\mathbf{T}}}(t\mu_0/\|\mu_0\|_{\infty}) : |t| < 1\}$ has at the origin zero of order $m \geq 1$ , i.e., $$h_{\mu_0}(t) := h \circ \phi_{\widetilde{\mathbf{T}}}(t\mu_0/\|\mu_0\|_{\infty}) = c_m t^m + c_{m+1} t^{m+1} + \dots,$$ then the growth of |h| on this disk is estimated by $$|h_{\mu_0}(t)| \le |t|^m (|t| + |c_m|)/(1 + |c_m||t|)$$ $$= \tanh d_{\widetilde{\mathbf{T}}}\left(\mathbf{0}, \phi_{\widetilde{\mathbf{T}}}\left(|t|^m \frac{|t| + |c_m|}{1 + |c_m||t|} \frac{\mu_0}{\|\mu_0\|_{\infty}}\right)\right) \le \tanh d_{\widetilde{\mathbf{T}}}\left(\mathbf{0}, \phi_{\widetilde{\mathbf{T}}}\left(t^m \frac{\mu_0}{\|\mu_0\|_{\infty}}\right)\right).$$ (3.4) The equality in the right inequality occurs (even for one $t_0 \neq 0$ ) only when $|c_p| = 1$ ; then $h_{\mu_0}(t)$ is a hyperbolic isometry of the unit disk and all terms in (3.4) are equal. Golusin's version mentioned above asserts that a holomorphic function $$g(t) = c_m t^m + c_{m+1} t^{m+1} + \dots : \Delta \to \Delta \quad (c_m \neq 0, \ m \ge 1)$$ is estimated in $\Delta$ by $$|g(t)| \le |t|^m \frac{|t| + |c_m|}{1 + |c_m||t|},$$ and the equality occurs only for $g_0(t) = t^m(t + c_m)/(1 + \overline{c}_m t)$ (see [Go, Ch. 8]). On the other hand, it follows from Theorem A and weak\* compactness of the closure of $\widetilde{\mathbf{T}}$ in $\mathbf{B} \times \Delta$ that for any fixed $t_0 \neq 0$ there is a holomorphic map $j(\varphi) : \widetilde{\mathbf{T}} \to \Delta$ such that $$d_{\Delta}(0,j\circ\phi_{\widetilde{\mathbf{T}}}(t_0\mu_0^))=c_{\widetilde{\mathbf{T}}}(\mathbf{0},\phi_{\widetilde{\mathbf{T}}}(t_0\mu_0^))=d_{\widetilde{\mathbf{T}}}(\mathbf{0},\phi_{\widetilde{\mathbf{T}}}(t_0\mu_0^*)).$$ where $\mu_0^* = \mu_0/\|\mu_0\|_{\infty}$ . Thus, letting $$\eta(t) = |t|^m (|t| + |c_m|)/(1 + |c_m||t|) \le |t|,$$ one derives $$|h_{\mu_0}(t_0)| \leq j \circ \phi_{\widetilde{\mathbf{T}}}(\eta(t_0)\mu_0^) = \tanh d_{\widetilde{\mathbf{T}}}(\mathbf{0}, \phi_{\widetilde{\mathbf{T}}}(\eta(t_0)\mu_0^)) \leq \tanh d_{\widetilde{\mathbf{T}}}(\mathbf{0}, \phi_{\widetilde{\mathbf{T}}}(|t_0|\mu_0^*))$$ which implies (3.4) (for details see [Kr9]). There is also a differential analog of the inequalities (3.4) which involves the infinitesimal metrics $\mathcal{C}_{\widetilde{\mathbf{T}}}$ and $\mathcal{K}_{\widetilde{\mathbf{T}}}$ . It will not be used here. 3.2. A holomorphic homotopy of univalent function. Similar to the functions in S, we define for each $F \in \Sigma$ with expansion (1.1) the complex homotopy $$F_t(z) = tF\left(\frac{z}{t}\right) = z + b_0 t + b_1 t^2 z^{-1} + b_2 t^3 z^{-2} + \dots : \Delta^* \times \Delta \to \widehat{\mathbb{C}}$$ (3.5) so that $F_0(z) \equiv z$ . Then $S_{F_t}(z) = t^{-2}S_F(t^{-1}z)$ , and moreover, this point-wise map determines a holomorphic map $$h_F(t) = S_{F_t}(\cdot): \Delta \to \mathbf{B}$$ (3.6) (see, e.g. [Kr3]). This map generates the homotopy disks $\Delta(S_F) = h_F(\Delta)$ of F in the space $\mathbf{T}$ and its covers in $\mathbf{T}_1$ defined by $$\Delta_1(S_F, b_0) := \{(S_F, tb_0) : |t| < 1\}.$$ These disks are holomorphic at noncritical points of map (3.6) and foliate both spaces and the set $\Sigma^0$ . The dilatations of the homotopy maps are estimated by

Proposition 3.2. [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 $$F_{b_0,b_1}(z) = z + b_0 + b_1 z^{-1}, \quad |b_1| = 1,$$ (3.7) whose homotopy maps $$F_{b_0,b_1t^2}(z) = z + b_0t + b_1t^2z^{-1} (3.8)$$ 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 [KK], $$k(F_t) = \frac{m+1}{2} |b_m| |t|^{m+1} + O(t^{m+2}), \quad t \to 0.$$ (3.9) This bound is sharp; it holds for the maps $$F_{m,t}(z) = \frac{1}{\kappa_{m,t}(1/z)} = z \left(1 - \frac{t}{z^{m+1}}\right)^{2/(m+1)} = z - \frac{2t}{m+1} \frac{1}{z^m} + \dots, \quad |t| \le 1, \quad (3.10)$$ whose extremal extension to $\mathbb C$ has Beltrami coefficient $\mu_{F_{m,t}}(z)=t|z|^{m-1}/z^{m-1}$ for |z|<1. The Teichmüller geodesic (extremal) disks $$\Delta(\psi) = \{ \phi_{\widetilde{\mathbf{T}}}(t\mu_0) : \ t \in \Delta \}$$ in the spaces T and T₁ are generated by $F \in \Sigma^0$ having extremal extensions with Beltrami coefficients $\mu_t(z) = t|\psi(z)|/\psi(z)$ , where $\psi$ is a holomorphic integrable quadratic differential on $\Delta$ and $\Delta \setminus \{0\}$ , respectively. Such an extension is unique (up to a constant factor of $\psi$ ) in their equivalence classes. In particular, any homotopy function $F_t$ has such an extremal extension. The Teichmüller disks foliate dense subsets in T and T₁ (and in $\Sigma^0$ ); cf. [GL], [St].

Lemma 3.3. [Kr7] Let $\lambda(|t|)d|t|$ be a circularly symmetric subharmonic metric on $\Delta$ such that $$\lambda(r) = mcr^{m-1} + O(r^m)$$ as $r \to 0$ with $0 < c \le 1$ $(m = 1, 2, ...),$ (3.12) and this metric has curvature at most -4 in the potential sense. Then $$\lambda(r) \ge \frac{mcr^{m-1}}{1 - c^2r^{2m}}.$$ Note that all metrics subject to (3.12) are dominated by $\lambda_m(t) = m|t|^{m-1}/(1-|t|^{2m})$ .

Lemma 4.1. For any $$F(z) = z + b_0 + b_1 z^{-1} + \dots \in \Sigma^0$$ , we have $$\max_{b_0} |\mathcal{J}^0(S_{F_t}, b_0 t)| = \max_{b_0} |\mathcal{J}^0(F_{b_0, b_1 t^2})| + O(t^{d+1}) \ge \max_{b_0} |\beta_{d/2}(b_0)| |b_1|^{d/2} |t|^d + O(t^{d+1}). \quad (4.10)$$ again taking the maximum over admissible $b_0$ . Proof. The homotopy disk of any function $F_{b_0,b_1}$ is extremal and admits the rotational symmetry. Accordingly, $|\mathcal{J}(F_{b_0,b_1t^2})|$ is circularly symmetric on the disk $\Delta$ . Define by (4.3) the corresponding functions $$g_{m,b_1}(t) = \mathcal{J}^0(S_{F_{b_0,b_1t^2}}, -F_{b_0,b_1t^2}(z_m))$$ and conformal metrics $$\lambda_{g_{m,b_1}}(t) = g_{m,b_1}^* \lambda_{\Delta}(t) = \frac{|g'_{m,b_1}(t)|}{1 - |g_{m,b_1}(t)|^2}$$ (whose Gaussian curvature equals -4 at noncritical points) and take the upper envelopes $$\mathcal{J}^{0}(t) := \sup_{m} |g_{m,b_{1}}(t)|, \quad \lambda_{\mathcal{J}^{0}}(t) := \sup_{m} \lambda_{g_{m,b_{1}}}(t).$$ Both envelopes are circularly symmetric continuous and subharmonic on $\Delta$ . Note also that (cf. (4.4)), $$\mathcal{J}^{0}(t) \leq |\mathcal{J}^{0}(S_{F_{t}}, b_{0}t)| = \sup_{m} |g_{m,b_{1}}(S_{F_{0,b_{1}t^{2}}})| = |\beta_{d/2}(b_{0})| |b_{1}|^{d/2} |t|^{d} + O(t^{d+1}). \tag{4.11}$$ Since $$\tanh^{-1}|g_{m,b_1}(r)| = \int_{0}^{|g_{m,b_1}(r)|} \frac{|dt|}{1 - |t|^2} = \int_{0}^{r} \lambda_{g_{m,b_1}}(t)|dt|,$$ we have $$\tanh^{-1}|\mathcal{J}^{0}(r)| = \sup_{m} \int_{0}^{r} \lambda_{g_{m,b_{1}}}(t)|dt| = \int_{0}^{r} \sup_{m} \lambda_{g_{m,b_{1}}}(t)|dt| = \int_{0}^{r} \lambda_{\mathcal{J}^{0}}(t)dt.$$ (4.12) The second equality in (4.12) is obtained by taking a monotone increasing subsequence of metrics $$\lambda_1 = \lambda_{g_{1,b_1}}, \ \lambda_2 = \max(\lambda_{g_{1,b_1}}, \lambda_{g_{2,b_1}}), \ \lambda_3 = \max(\lambda_{g_{1,b_1}}, \lambda_{g_{2,b_1}}, \lambda_{g_{3,b_1}}), \ \dots$$ so that $$\lim_{p \to \infty} \lambda_p(t) = \sup_{m} \lambda_{g_{m,b_1}}(t).$$ Combining (4.12) with (4.4) and (4.7), one gets for small r, $$\tanh^{-1}[\mathcal{J}^0(S_{F_1}, b_0 r)] = \tanh^{-1}[\mathcal{J}^0(r)] + O(r^{d+1}) = \int_0^r \lambda_{\mathcal{J}^0}(t) dt + O(r^{d+1}). \tag{4.13}$$ Now observe that the metric $\lambda_{\mathcal{J}^0}$ has in a neighborhood of any $t_0 \in \Delta$ a supporting metric of curvature -4, and therefore its curvature in $\Delta$ in the potential sense is at most -4. Thus this metric can be estimated from below by Lemma 3.3 (with m=d) which implies the lower bound $$\lambda_{\mathcal{J}^0}(r) \ge \frac{dCdr^{d-1}}{1 - C^2r^{2d}}, \quad r < 1,$$ (4.14) where $$C = \max_{b_0} |\beta_{d/2}(b_0)| |b_1|^{d/2}.$$ Integrating (4.14) over a small radial segment [0, r], one obtains $$\int_{0}^{r} \lambda_{\mathcal{J}^{0}}(t)dt \ge \tanh^{-1}(Cr^{d}) + O(r^{d+1}, \quad r \to 0,$$ which provides after substitution into (4.13) the desired estimate (4.11). Comparison of the relations (4.7) and (4.9)-(4.11) yields $$r^{d}|\mathcal{J}^{0}(S_{F}, b_{0})| = |\mathcal{J}^{0}(S_{F_{r}}, b_{0}r)| = \max_{b_{0}} |\widetilde{J}^{0}(F_{b_{0}, b_{1}r^{2}})| + O(r^{d+1})$$ $$= \max_{b_{0}} |\beta_{d/2}(b_{0})| |b_{1}|^{d/2} r^{d} + O(r^{d+1}), \quad r \to 0,$$ and letting $r \to 0$ , $$|\mathcal{J}^{0}(S_{F}, b_{0})| = \max_{b_{0}} |\widetilde{J}^{0}(F_{b_{0}, b_{1}})| = \max_{b_{0}} |\beta_{d/2}(b_{0})| |b_{1}|^{d/2}$$ $$= \max_{b_{0}} |\beta_{d/2}(b_{0})| \left(\frac{|S_{f}(0)|}{6}\right)^{d/2} \leq 1.$$ (4.15) This estimate is established for all $f \in S$ . Since for any extremal function $f_0(z) = z + \sum_{n=0}^{\infty} a_n^0 z^n$ of the functional J and its inversion $F_0(z) = F_{f_0}(z) = z + b_0^0 + b_1^0 z^{-1} + \dots$ obeying (2.3) must be $$\frac{|\widetilde{J}(S_{F_0}, -a_2^0)|}{M(J)} = |\mathcal{J}^0(S_{F_0}, -a_2^0)| = 1$$ and $|\beta_{d/2}(b_0)| \leq 1$ , it follows from (4.15) that necessarily $\max_{b_0} |\beta_{d/2}(b_0)| = 1$ and $$|b_1^0| = \frac{1}{6}|S_{f_0}(0)| = |(a_2^0)^2 - a_3^0| = 1.$$ 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 (1.7). In addition, the extremality of $f_0$ implies $$|J(f_0)| = M(J) = \max\{|J(\kappa_{\theta})|, |J(\kappa_{2,\theta})|\}.$$ - 3⁰. 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 4.2. 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 $z_j \in \Gamma$ with $\alpha_j = 0$ or $\alpha_j \le l$ , 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 4.2 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. $\mathbf{4}^{0}$ . Finally, consider the case when J has no extremals $f_{0}$ satisfying (2.3), and hence any extremal inversion $F_{f_{0}}$ is of the form $$F(z) = z + b_0 + b_m z^{-m} + b_{m+1} z^{-(m+1)} + \dots (b_m \neq 0; |z| > 1) (4.16)$$ 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. One can apply to $J^{m+1}$ the above arguments, replacing $F_{0,b_1t^2}$ by the corresponding function $F_{m,t}(z) + b_0$ , where $F_{m,t}$ is given by (3.10) and $b_0$ is the same as in (4.16). Its Schwarzian relates to $S_{F_t}$ by $S_{F_t} = S_{F_{m,t}} + O(t^{m+1})$ as $t \to 0$ . Now $$[J(S_{F_{m,t}}, b_0 t)/M(J)]^{m+1} = \beta_d(b_0) t^{d(m+1)} + \dots,$$ and after applying the asymptotic estimate (3.9), one obtains instead of (4.15) the bound $$\left| \frac{\mathcal{J}(S_F, b_0)}{M(J)} \right|^{m+1} \le \max_{b_0} \left| \beta_d(b_0) \right| \left( \frac{m+1}{2} |b_m| \right)^d \le 1,$$ or $$\left| \frac{\mathcal{J}(S_F, b_0)}{M(J)} \right| \le \max_{b_0} \left| \beta_d(b_0) \right|^{1/(m+1)} \left( \frac{m+1}{2} |b_m| \right)^{d/(m+1)} \le 1. \tag{4.17}$$ In the case of an extremal function $F_{f_0}(z) = z + b_0^0 + b_m^0 z^{-m} + \dots$ for $\widetilde{J}(F)$ , it must be $|\mathcal{J}(S_{F_{f_0}})/M(J)| = 1$ , and (4.17) implies $$\frac{m+1}{2}|b_m^0| = 1.$$ Since the functions (4.16) with m > 1 satisfy $b_1 = \cdots = b_{m-1} = 0$ , one can apply the well-known coefficient estimates of Golusin and Jenkins (see [Go, Ch. XI], [Je]) which provide in our case the bound $$|b_m| \le 2/(m+1) \tag{4.18}$$ with equality only for $F = F_{m,t}$ with |t| = 1 (up to translation $F_{m,t}(z) + c$ ). In this case, $M(J) = |J(\kappa_{m,\theta})|$ . We have established that any extremal function $f_0$ maximizing |J(f)| must be of the form (2.1). The theorem is proved.

Proposition 5.1. [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 $$|a_n| \le 2k/(n-1),\tag{5.2}$$ with equality only for the functions $$f_{n-1,t}(z) = f_{1,t}(z^{n-1})^{1/(n-1)} = z + \frac{2kt}{n-1}z^n + \dots, \quad n = 3, 4, \dots$$ (5.3) Note that every function (5.3) admits a quasiconformal extension $\widehat{f}_{n-1,t}$ onto $\Delta^*$ with Beltrami coefficient $\mu_n(z) = t|z|^{n+1}/z^{n+1}$ and $\widehat{f}_{n-1,t}(\infty) = \infty$ . Accordingly, $\widehat{F}_{n-1,t}(z) = 1/\widehat{f}_{n-1,t}(1/z) \in \Sigma^0$ admits a quasiconformal extension onto the unit disk with $\widehat{F}_{n-1,t}(0) = 0$ and $\mu_{\widehat{F}_{n-1,t}}(z) = t|z|^{n-1}/z^{n-1}$ for |z| < 1. Another essential point is that for any function $$F_{n-1}(z) := \widehat{F}_{n-1,1}(z) = 1/\kappa_0 (1/z^{n-1})^{1/(n-1)},$$ its homotopy disk $\{S_{F_{n-1}}\}$ in T is Teichmüller geodesic. Together with estimate (5.2), this implies that for any m > 2 and small r > 0, $$|J_n(\kappa_{m,r})| < r(n-1)^2;$$ thus $$|J_n(\kappa_{m,\theta})| < (n-1)^2 = J_n(\kappa_0),$$ completing the proof of Theorem 2.2.

Theorem 6.1. 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 (3.1) 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 and $I_n^p(f) = |a_n^p - a_2^{p(n-1)}|$ satisfy the assumptions of Theorem 2.1. The same arguments as in the proof of Theorem 2.2 imply $$|I_n^p(\kappa_{m,\theta})| < |I_n^p(\kappa_{\theta})|$$ for all $m > 2$ , completing the proof. In the same way, one obtains

Theorem 6.2. 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 $f = \kappa_{\theta}$ .

Theorem 7.1. Let $\{J_j(f)\}_1^{\infty}$ be a sequence of uniformly bounded homogeneous holomorphic (not necessarily distinct) functionals on S of degrees $d_j = d(J_j)$ satisfying the assumptions of Theorem 2.1 and such that $$|J_j(\kappa_{m,\theta})| \le |J_j(\kappa_0)| \quad \text{for all } m > 1. \tag{7.1}$$ Then for any $f \in S^0$ , $$\lim_{j \to \infty} \sup \left| \frac{J_j(f)}{J_j(\kappa_0)} \right| = v(f) < 1.$$ (7.2) A similar result holds for the upper envelop $\sup_{\alpha} |J_{\alpha}(f)|$ of a family of uniformly bounded homogeneous holomorphic functionals on S. Proof. It follows from (7.1) and Theorem 2.1 that only the Koebe function $\kappa_{\theta}$ is extremal for each $J_j$ . We lift $J_j(f)$ to holomorphic functionals $\widehat{J}_j(\varphi)$ on the space $\mathbf{T}_1$ (taking again $\varphi = (S_f, -a_2(f))$ ) and consider the ratios $$v(\varphi) = \limsup_{j \to \infty} \left| \frac{\widehat{J}_j(\varphi)}{\widehat{J}_j(\varphi_0)} \right|$$ where $\varphi_0 = (S_{\kappa_0}, -2)$ . The function $v(\varphi)$ is well defined on this space and $v(\varphi) \leq 1$ . Its upper semicontinued regularization $v^*(\varphi) = \limsup_{\varphi' \to \varphi} v(\varphi') \leq 1$ is plurisubharmonic on $\mathbf{T}_1$ , hence by the maximum principle it cannot attain the value 1 inside $\mathbf{T}_1$ ; otherwise this function must be identically equal to 1. But, for example, $v^*(\mathbf{0}) = v(\mathbf{0}) = 0$ , since near the origin by Schwarz's lemma, $$\left| \frac{\widehat{J}_{j}(\varphi)}{\widehat{J}_{j}(\varphi_{0})} \right| \leq \operatorname{const} \|\varphi\|$$ for all p. Therefore, $v(\varphi) \leq v^*(\varphi) < 1$ , which completes the proof of (7.2). In the case of Zalcman's functional this yields that for any $f \in S^0$ , $$v(f) = \limsup_{n \to \infty} \frac{|a_n^2 - a_{2n-1}|}{(n-1)^2} < 1$$ (cf. [Ha], [EV]). On the other hand, the bound (1.5) implies that $v(f) \leq 1$ for any $f \in S$ . Similar estimates hold for perturbations of this functional via (5.4).