Teichmuller space theory and classical problems of geometric function theory

Theorem 1. Any rotationally invariant polynomial functional (4), whose zero set $\mathcal{Z}_J = \{w \in \widehat{S} : J(w) = 0\}$ is separated from the set (3), is maximized on the class $\mathcal{X}$ only by functions $w_{0,\tau,\theta} \in \mathcal{R}_{\mathcal{X}}$ . In other words, any extremal function $w_0$ of any homogeneous (rotationally invariant) coefficient functional J on a rotationally invariant and variationally stable class $\mathcal{X}$ must be simultaneously maximal for the second coefficient $a_2$ on this class, unless $J(w_0) = 0$ . All assumptions of this theorem on the class $\mathcal{X}$ and the functional J are essential and cannot be omitted. This will be illustrated on examples in the last section. 1.3. The following two classes of univalent functions are of special interest. First, let $\mathcal{X}$ be the canonical class S of univalent functions $w(z) = z + \sum_{n=1}^{\infty} a_n z^n$ on $\mathbb{D}$ with w(0) = 0, w'(0) = 1. The classical result for this class states $|a_2| \leq 2$ , with equality only for the Koebe function $$\kappa_0(z) = \frac{z}{(1-z)^2} = z + \sum_{n=0}^{\infty} nz^n$$ (5) mapping the unit disk onto the complement of the ray $\{w = -t : 1/4 \le t \le \infty\}$ , and for the rotations $\kappa_{\theta}(z) = e^{-i\theta} \kappa_0(e^{i\theta}z)$ of this function. In this case, Theorem 1 implies an alternate and direct proof of de Branges theorem solving the Bieberbach conjecture that $|a_n| \leq n$ for all $f \in S$ (see [3], [8], [12]), and moreover, Theorem 1 yields that the Koebe function (5) is extremal for all coefficient functionals of type (4). The second case concerns the collections $\mathcal{X}(\Gamma)$ of univalent functions w(z) on $\mathbb{D}$ compatible with the hyperbolic Fuchsian groups $\Gamma$ acting on $\mathbb{D}$ . This means that the maps $$w_{\gamma} = w \circ \gamma \circ w^{-1}, \quad \gamma \in \Gamma,$$ are the Moebius transformations of the sphere $\widehat{\mathbb{C}}$ , and the group $\Gamma' = w\Gamma w^{-1}$ is a discrete subgroup the Moebius group $\operatorname{Mob}(\widehat{\mathbb{C}})$ . As a consequence of Theorem 1, one obtains for such collections as the following result.

Theorem 2. (i) For every Fuchsian group $\Gamma$ and any homogeneous polynomial functional (1) on $\mathcal{X}(\Gamma) \subset \mathbf{T}$ satisfying the assumptions of Theorem 1, any its maximizing function $w_0(z)$ must simultaneously maximize the second coefficient $a_2(w)$ on this class. The rotations connecting the extremal functions of the functional (4) correspond to conjugation of both groups $\Gamma$ and $w_0\Gamma w_0^{-1}$ by elements of $\operatorname{Mob}(\widehat{\mathbb{C}})$ . This theorem opens ways to investigations of extremal problems for quasiconformal deformations of Fuchsian groups. One of the interesting open questions here is, for which groups $\Gamma$ and extremals $w_0$ , the group $w_0\Gamma w_0^{-1}$ is a totally degenerated functional group, i.e., such that its set of discontinuity $\Omega(w_0\Gamma w_0^{-1})$ is a simply connected domain (dense in $\widehat{\mathbb{C}}$ ). 1.4. As was mentioned above, the proof of Theorem 1 involves the deep results of Teichmüller space theory. The functional J is lifted from $\mathcal{X}$ to the the corresponding submanifold $\mathcal{X}_{\mathbf{T}}$ in the Teichmüller space $\mathbf{T}_1$ of the punctured disk $\mathbb{D}_* = \{0 < |z| < 1\}$ . This space is biholomorphically equivalent to the Bers fiber space $\mathcal{F}(\mathbf{T})$ over the universal Teichmüller space $\mathbf{T} = \text{Teich}(\mathbb{D})$ (see [2]). This generates a holomorphic functional $\mathcal{J}(\varphi, t)$ on the image of $\mathcal{X}_{\mathbf{T}}$ in $\mathcal{F}(\mathbf{T})$ covering J, with the same range domain as the initial finctional J. Here $\varphi = S_w \in \mathcal{X}$ are the Schwarzian derivatives of the initial univalent functions, while the second variable t runs over the fiber domain $w_{\varphi}(\mathbb{D})$ defined by $\varphi$ . A crucial step in the proof is to maximize $|\mathcal{J}(\varphi,t)|$ over $\varphi$ by a fixed t.

Lemma 1. For any $\mu \in \text{Belt}(\mathbb{D}^*)_1$ and any $\theta \in [0, 2\pi)$ , 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 $$w(0) = 0, \quad w'(0) = e^{i\theta}, \quad w(1) = 1.$$ (7) This solution is holomorphic on the unit disk $\mathbb{D}$ , and hence, $w(z_0) = \infty$ at some point $z_0$ with $|z_0| \ge 1$ (so w(z) does not have a pole in $\mathbb{D}$ ). Proof. Let us first consider the coefficients $\mu$ vanishing in a broader disk $\mathbb{D}_r = \{|z| < r\}, r > 1$ , so that $w^{\mu}$ is conformal on $\mathbb{D}_r \ni \overline{\mathbb{D}}$ , and assume that $\mu \neq \mathbf{0}$ (the origin of Belt( $\mathbb{D}^*$ )₁). Fix $a \in (1, r)$ close to 1 and $\theta \in [0, 2\pi]$ ; then $1/a \in \mathbb{D}$ . The generalized Riemann mapping theorem for the Beltrami equation $\partial_{\overline{z}}w = \mu(z)\partial_z w$ on $\widehat{\mathbb{C}}$ implies a homeomorphic solution $\widehat{w}$ to this equation satisfying $$\widehat{w}(-1/a) = -1/a, \quad \widehat{w}'(-1/a) = e^{i\theta}, \quad \widehat{w}(\infty) = \infty.$$ (8) Its composition with the Möbius map $$\gamma_a(z) = (1 - az)/(z - a)$$ preserving either from disks $\mathbb{D}$ and $\mathbb{D}^*$ has the Beltrami coefficient $$\gamma_{a,*}\mu := \mu_{\widehat{w} \circ \gamma_a}(z) = \mu \circ \gamma_a(z) \gamma_a'(z) / \overline{\gamma_a'(z)}$$ and also is conformal in the disk $\mathbb{D}_r$ and holomorphic in $\mathbb{D}$ . Since, by the classical Schwarz lemma, for any holomorphic map $g: \mathbb{D} \to \mathbb{D}$ and any point $z_0 \in \mathbb{D}$ , $$|g'(z_0)| \le (1 - |g(z_0)|^2)/(1 - |z_0|^2)$$ with equality only for appropriate Möbius automorphism of $\mathbb{D}$ , the above normalization (8) and the assumption on $\mu$ yield that for any map $\widehat{w}(z)$ the image $\widehat{w}(\mathbb{D})$ does not cover $\mathbb{D}$ , and thus either $\widehat{w}(\mathbb{D})$ is a proper subdomain of $\mathbb{D}$ or it also contains the points z with |z| > 1 outer for $\mathbb{D}$ . Passing if needed to suitable rotated map $e^{-i\alpha}\widehat{w}(e^{i\alpha}z)$ , one obtains that this domain $\widehat{w}(\mathbb{D})$ does not contain simultaneously both distinguished points a and 1/a (at least sufficiently close to 1). Now consider the map $$w_{a,a}(z) = \gamma_a^{-1} \circ \widehat{w} \circ \gamma_a(z),$$ having the same Beltrami coefficient $\gamma_{a,*}\mu$ . Since $$\gamma_a(\infty) = -a, \quad \gamma_a(a) = \infty, \quad \gamma_a(0) = -1/a, \quad \gamma_a(1/a) = 0$$ (and accordingly, $\gamma_a^{-1}(\infty) = a$ , $\gamma_a^{-1}(-a) = \infty$ , $\gamma_a^{-1}(0) = 1/a$ , $\gamma_a^{-1}(-1/a) = 0$ ), the map $w_{a,a}$ satisfies $$w_{a,a}(0) = 0, \quad w'_{a,a}(0) = \widehat{w}'(-1/a) = e^{i\theta}, \quad w_{a,a}(a) = \gamma_a^{-1} \circ w \circ \gamma_a(a) = a.$$ (9) If one starts with Beltrami coefficient $\gamma_{a,*}^{-1}\mu$ , taking its map w normalized by (7), then the final map $w_{a,a}$ has the initial Beltrami coefficient $\mu$ and satisfies (8) for all $a \in (1,r)$ sufficiently close to 1. In view of our assumptions on $\widehat{w}$ , the point $\widehat{w}^{-1}(a)$ cannot lie in the unit disk $\mathbb{D}$ ; therefore the function $w_{a,a}$ is holomorphic in this disk. Now we investigate the limit process as $a \to 1$ . Any from the constructed maps $w_{a,a}$ is represented as a composition of a fixed solution $\widehat{w}$ to the equation $\partial_{\overline{z}}w = \mu(z)\partial_z w$ subject to (10) and some Möbius maps $\widehat{\gamma}_a$ . The first two conditions in (9) imply that the restrictions of these $\widehat{\gamma}_a$ to $\widehat{w}(\mathbb{D}_r)$ form a (sequentially) compact set of $\widehat{\gamma}_a$ in the topology of convergence in the spherical metric on $\widehat{\mathbb{C}}$ . Letting $a \to 1$ , one obtains in the limit the map $\widehat{\gamma}_1(z) = \lim_{a \to 1} \widehat{\gamma}_a(z)$ , which also is a non-degenerate (nonconstant) Möbius map. Accordingly, $$\lim_{a \to 1} w_{a,a}(z) = \widehat{\gamma}_1 \circ \widehat{w}(z) =: \widehat{w}_1(z),$$ and this map satisfies (9) with a = 1, which is equivalent to (7). Note that the relations (9) do not depend on r and that the normalization (8) also is valid for the inverse rotation $e^{i\alpha}\widehat{w}_1(e^{-i\alpha}z)$ of the limit function. This implies the assertion of Lemma 2 for all Beltrami coefficients $\mu \neq \mathbf{0}$ supported in the disk $\mathbb{D}_r^* = \{|z| > r\}$ with r > 1. To extend the obtained result to arbitrary $\mu \in \text{Belt}(\mathbb{D}^*)_1$ ( $\mu \neq \mathbf{0}$ ), we pass to coefficients $\mu_r(z) = \mu(z)$ for |z| > r and extended by zero to $\mathbb{D}_r$ . The compactness properties of the k-quasiconformal families (i.e., with $\|\mu\|_{\infty} \leq k < 1$ ) imply the convergence of maps $w^{\mu_r}(z)$ normalized by (7) to $w^{\mu}(z)$ as $r \to 1$ in the spherical metric on $\widehat{\mathbb{C}}$ (and hence everywhere on $\widehat{\mathbb{C}}$ ). The remained case $\mu(z) \equiv 0$ omitted above follows in the limit as $\mu \to \mathbf{0}$ (or even $\mu(z) \to 0$ almost everywhere in $\mathbb{D}^*$ ). Then the map w(z) satisfying (7) must be an elliptic fractional linear transformation with fixed points 0 and 1; hence, $$\frac{w-1}{w} = e^{i\theta} \, \frac{z-1}{z},$$ which implies $$w = \frac{e^{-i\theta}z}{(e^{-i\theta} - 1)z + 1}. (10)$$ The simple direct calculations yield that $w(z_0) = \infty$ only at a point $z_0$ with $|z_0| \ge 1$ (which also follows from the above). The proof of Lemma 2 is completed. This lemma plays an important role in our further considerations. We shall consider the univalent function w(z) in the disk $\mathbb{D}$ normalized by (9) and their rotations (1) with $\tau, \theta \in [0, 2\pi]$ ; all these rotations also are holomorphic univalent in this disk, so $w_{\tau,\theta}(z_0) = \infty$ only at some point $z_0 \in \overline{\mathbb{D}^*}$ .

Lemma 2. Let w(z) be a quasiconformal map of the plane $\widehat{\mathbb{C}}$ with Beltrami coefficient $\mu(z)$ which satisfies $\|\mu\|_{\infty} < \varepsilon_0 < 1$ and vanishes in the disk $\{|z| < r\}$ . Suppose that w(0) = 0, w'(0) = 1, and w(1) = 1. Then, for sufficiently small $\varepsilon_0$ and for $|z| \le R < r_0(\varepsilon_0, r)$ we have the variational formula $$w(z) = z - \frac{z^2(z-1)}{\pi} \iint_{|\zeta| > r} \frac{\mu(\zeta)d\xi d\eta}{\zeta^2(\zeta-1)(\zeta-z)} + \Omega_{\mu}(z),$$ where $\zeta = \xi + i\eta$ ; $\max_{|z| \leq R} |\Omega_{\mu}(z)| \leq C(\varepsilon_0, r, R) \|\mu\|_{\infty}^2$ ; $r_0(\varepsilon_0, r)$ is a well defined function of $\varepsilon_0$ and r such that $\lim_{\varepsilon_0 \to 0} r_0(\varepsilon_0, r) = \infty$ , and the constant $C(\varepsilon_0, r, R)$ depends only on $\varepsilon_0$ , r and R. Step 2: Lifting to covering space $T_1$ and estimating the restricted plurisubharmonic functional. Our next step is to lift both polynomial functionals J(w) and $\widehat{J}(W)$ onto the Teichmüller space $T_1$ which covers T. Letting $$\widehat{J}(\mu) = \widetilde{J}(W^{\mu}), \tag{12}$$ we lift these functionals from the sets $S_{\theta}(1)$ and $\Sigma_{\theta}(1)$ onto the ball $Belt(\mathbb{D})_1$ . Then, under the indicated $\mathbf{T}_1$ -equivalence, i.e., by the quotient map $$\phi_{\mathbf{T}_1}: \operatorname{Belt}(\mathbb{D})_1 \to \mathbf{T}_1, \quad \mu \to [\mu]_{\mathbf{T}_1},$$ the functional $\widetilde{J}(W^{\mu})$ is pushed down to a bounded holomorphic functional $\mathcal{J}$ on the space $\mathbf{T}_1$ with the same range domain. Equivalently, one can apply the quotient map $\operatorname{Belt}(\mathbb{D})_1 \to \mathbf{T}$ (i.e., T-equivalence) and compose the descended functional on T with the natural holomorphic map $\iota_1: \mathbf{T}_1 \to \mathbf{T}$ generated by the inclusion $\mathbb{D}_* \to \mathbb{D}$ forgetting the puncture. Note that since the coefficients $b_0, b_1, \ldots$ of $W^{\mu} \in \Sigma_{\theta}$ are uniquely determined by its Schwarzian $S_{W^{\mu}}$ , the values of $\mathcal{J}$ in the points $X_1, X_2 \in \mathbf{T}_1$ with $\iota_1(X_1) = \iota_1(X_2)$ are equal. Now, using the Bers isomorphism theorem, we regard the points of the space $\mathbf{T}_1$ as the pairs $X_{W^{\mu}} = (S_{W^{\mu}}, W^{\mu}(0))$ , where $\mu \in \text{Belt}(\mathbb{D})_1$ obey $\mathbf{T}_1$ -equivalence (hence, also $\mathbf{T}$ -equivalence). Denote (for simplicity of notations) the composition of $\mathcal{J}$ with biholomorphism $\mathbf{T}_1 \cong \mathcal{F}(\mathbf{T})$ again by $\mathcal{J}$ . In view of (5) and (13), it is presented on the fiber space $\mathcal{F}(\mathbf{T})$ by $$\mathcal{J}(X_{W^{\mu}}) = \mathcal{J}(S_{W^{\mu}}, t), \quad t = W^{\mu}(0). \tag{13}$$ This yields a logarithmically plurisubharmonic functional $|\mathcal{J}(S_{W^{\mu}},t)|$ on $\mathcal{F}(\mathbf{T})$ . Note that since the coefficients $b_0, b_1, \ldots$ of $W^{\mu} \in \Sigma_{\theta}$ are uniquely determined by its Schwarzian $S_{W^{\mu}}$ , the values of $\mathcal{J}$ in the points $X_1, X_2 \in \mathbf{T}_1$ with $\iota_1(X_1) = \iota_1(X_2)$ are equal. We have to estimate a smaller plurisubharmonic functional arising after restriction of $\mathcal{J}(S_{W^{\mu}}, t)$ to $S_W \in \mathcal{X}_{\mathbf{T}}$ , i.e., the restriction of functional (13) onto the corresponding set of pairs $(S_{W^{\mu}}, W^{\mu}(0))$ consisting of $S_{W^{\mu}} \in \mathcal{X}_{\mathbf{T}}$ and of the values $W^{\mu}(0)$ filling some subdomain $D_{\mathcal{X},\theta}$ . Since our functionals are polynomials, they are defined for all $S_W \in \mathbf{T}$ and t from some domain $D_{\theta}$ containing $D_{\chi,\theta}$ . We define on $D_{\theta}$ the function $$u_{\theta}(t) = \sup_{W^{\mu}} |\mathcal{J}(S_{W^{\mu}}, t)|, \tag{14}$$ where the supremum is taken over all $S_{W^{\mu}} \in \mathbf{T}$ admissible for a given $t = W^{\mu}(0) \in D_{\theta}$ . The following basic lemma from [13] provides that this function inherits subharmonicity of $\mathcal{J}$ .

Lemma 3. The function $u_{\theta}(t)$ is subharmonic in the domain $D_{\theta}$ . The proof of this lemma in [13] is complicated. It involves a weak approximation of the underlying space $\mathbf{T}$ (and simultaneously of the space $\mathbf{T}_1$ ) by finite dimensional Teichmüller spaces of the punctured spheres in the topology of locally uniform convergence on $\mathbb{C}$ and using the increasing unions of the quotient spaces $$\mathcal{T}_s = \bigcup_{j=1}^s \widehat{\Sigma}_{\theta_j}^0 / \sim = \bigcup_{j=1}^s \{ (S_{W_{\theta_j}}, W_{\theta}^{\mu}(0)) \} \simeq \mathbf{T}_1 \cup \dots \cup \mathbf{T}_1, \tag{15}$$ where $\theta_j$ run over a dense subset $\Theta \subset [-\pi, \pi]$ , the equivalence relation $\sim$ means $\mathbf{T}_1$ -\nequivalence on a dense subset $\widehat{\Sigma}^0(1)$ in the union $\widehat{\Sigma}(1)$ formed by univalent functions $W_{\theta_j}(z) = e^{-i\theta_j}z + b_0 + b_1z^{-2} + \ldots$ on $\mathbb{D}^*$ with quasiconformal extension to $\widehat{\mathbb{C}}$ satisfying $W_{\theta_j}(1) = 1$ , and $$\mathbf{W}_{\theta}^{\mu}(0) := (W_{\theta_1}^{\mu_1}(0), \dots, W_{\theta_s}^{\mu_s}(0)).$$ The Beltrami coefficients $\mu_j \in \text{Belt}(\mathbb{D})_1$ are chosen here independently. 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 $\mathcal{T}_s$ onto $\mathcal{F}(\mathbf{T})$ . The function (14) is determined by $$u(t) = \sup_{\Theta} u_{\theta_s}(t),$$ where $u_{\theta_s}$ is obtained by maximization of type (14) over $\mathcal{T}_s$ . Since, by assumption, the image $\mathcal{X}_{\mathbf{T}}$ of the class $\mathcal{X}$ in $\mathbf{T}$ is a complex manifold, the restriction of the function $|\mathcal{J}(S_{W^{\mu}},t)|$ to this manifold and to corresponding values of $t=W^{\mu}(0)$ also is plurisubharmonic. The arguments from [13] are straightforwardly extended to this restriction, giving in a similar way, that also the corresponding maximal function $$u_{\theta}(t) = \sup_{W^{\mu}} |\mathcal{J}(S_{W^{\mu}}, t)|$$ is subharmonic on the domain $D_{\mathcal{X},\theta}$ . Step 3: Range domain of $W^{\mu}(0)$ . The next step in maximization of the function $u_{\theta}$ (and thereby of the functional $\mathcal{J}$ ) is to establish the value domain of $W^{\mu}(0)$ for $W^{\mu}$ running over $\mathcal{X}_{\mathbf{T}}$ . This requires the corresponding covering estimate. Let 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 $\mathbf{T}$ modeled as a bounded subdomain of $\mathbf{B}$ . Consider in the unit disk the corresponding Schwarz differential equations $S_w(z) = \chi(\mathbf{x})$ and pick their holomorphic univalent solutions w(z) in $\mathbb{D}$ satisfying w(0) = 0, w'(0) = 1 (hence $w(z) = z + \sum_{n=0}^{\infty} a_n z^n$ ). Put $$|a_2^0| = \sup\{|a_2|: S_w \in \chi(G)\},\tag{16}$$ and let $w_0(z) = z + a_2^0 z^2 + \dots$ be one of the maximizing functions.

Lemma 4. (a) For every indicated solution $w(z) = z + a_2 z^2 + \ldots$ of the Schwarz differential equation, the image domain $w(\mathbb{D})$ covers entirely the disk $D_{1/(2|a_2^0|)} = \{|w| < 1/(2|a_2^0|)\}$ . The radius value $1/(2|a_2^0|)$ is sharp for this collection of functions, and the circle $\{|w| = 1/(2|a_2^0|) \text{ contains points not belonging to } w(\mathbb{D}) \text{ if and only if } |a_2| = |a_2^0| \text{ (i.e., when } w \text{ is one of the maximizing functions)}.$ (b) The inverted functions $$W(\zeta) = 1/w(1/\zeta) = \zeta - a_2^0 + b_1\zeta^{-1} + b_2\zeta^{-2} + \dots$$ map the disk $\mathbb{D}^*$ onto a domain whose boundary is entirely contained in the disk $\{|W + a_2^0| \le |a_2^0|\}$ . The proof follows the classical lines of Koebe's 1/4 theorem (cf. [6]). (a) Suppose that the point w = c does not belong to the image of $\mathbb{D}$ under the map w(z) defined above. Then $c \neq 0$ , and the function $$w_1(z) = cw(z)/(c - w(z)) = z + (a_2 + 1/c)z^2 + \dots$$ also belongs to this class, and hence by (16), $|a_2 + 1/c| \le |a_2^0|$ , which implies $$|c| \ge 1/(2|a_2^0|).$$ The equality holds only when $$|a_2 + 1/c| = |1/c| - |a_2| = |a_2^0|$$ and $|a_2| = |a_2^0|$ . (b) If a point $\zeta = c$ does not belong to the image $W(\mathbb{D}^*)$ , then the function $$W_1(z) = 1/[W(1/z) - c] = z + (c + a_2)z^2 + \dots$$ is holomorphic and univalent in the disk $\mathbb{D}$ , and therefore, $|c+a_2| \leq |a_2^0|$ . The lemma follows. This lemma implies that the boundary of the range domain of $W^{\mu}(0)$ is contained in the disk $$\mathbb{D}_{2|a_2^0|} = \{W: |W| < 2|a_2^0|\} \tag{17}$$ and the boundary of this domain touches from inside the circle $\{|W| = 2|a_2^0|\}$ at the points corresponding to extremal functions maximizing $|a_2|$ on the closure of the manifold $\mathcal{X}_T$ . Step 5. Finishing the proof. To complete the proof of the theorem, we have to apply some special variations of univalent functions with quasiconformal extension given by the following lemma, which is a special case of more general results from [9]. Here we essentially use the variational stability of $\mathcal{X}$ .

Lemma 5. 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 $\alpha_i = 0$ if $z_i \in E$ . 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 $\mu_h(z) = \partial_{\bar{z}}h/\partial_z h$ 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. We apply this lemma to quasiconformally extendable functions from $\mathcal{X}$ , which are dense in this subclass (in topology of locally uniform convergence on $\mathbb{D}$ ). Let $w_0$ be an extremal of a given functional J on $\mathcal{X}$ . Pass to the function $$w_{0r}(z) = \frac{1}{r}w_0(rz) = re^{i\theta}z + r^2a_2^0z^2 + \dots$$ with r close to 1, and to its image $\widetilde{w}_{0r}$ in $\widehat{S}(1)$ , using the corresponding rotations of type (1). This image is univalent and holomorphic on the closed disk $\overline{\mathbb{D}}$ and satisfy also the third normalization condition w(1) = 1. Varying appropriately the coefficients $a_2, a_{m_1}, \ldots, a_{m_s}$ of $\widetilde{w}_{0r}$ by Lemma 5 with quasiconformal variation h conformal on $\widetilde{w}_{0r}(\mathbb{D})$ , one derives that the maximal function $$u(t) = \sup_{\theta} u_{\theta}(t) = \sup \left\{ |\mathcal{J}(S_{W^{\mu}}, t)| : S(W^{\mu}) \in \bigcup_{s} \mathcal{T}_{s} \right\}$$ (18) (where $u_{\theta}$ and $\mathcal{T}_s$ are determined by (14) and (15)) must be positive on any circle $\{|t| = r < 2|a_2^0|\}$ , and hence on the whole disk (17). This function also is subharmonic and circularly invariant on this disk. Therefore, the maximal value of the function (18), which coincides with $\max |J(w)|$ on $\widehat{S}(1)$ , is attained on the boundary circle $\{|t|=2|a_2^0|\}$ and equals u(1). This is equivalent to assertion of Theorem 1.