A General Theorem on Sharp Evaluation of Polynomial Coefficients

Abstract

The evaluation of polynomial coefficients is an important problem of algebra, complex analysis and related fields of mathematics. This subject has a long history but is still actively investigated. The paper provides sharp estimates of polynomial coefficients by applying the intrinsic features of univalent functions and of Teichmüller spaces and solves the problem for all $n\ge 5$, giving the related extremal polynomial. Also, the case of nonvanishing polynomials in the disk, having an independent interest, is considered.

1. Introductory Remarks

Estimating the coefficients of polynomials and of holomorphic functions has a long history but still remains an important problem in many algebraic, geometric and physical applications of complex analysis. On the related investigations in geometric function theory, see, e.g., [1,2,3,4,5,6,7,8,9,10]. One of the interesting and still not sufficiently investigated directions is the quantitative investigation of polynomials as the points of Banach spaces.

Denote the collection of polynomials

$$p_n\left(z\right)=c_0+c_{1}z+\cdots +c_n{z}^n$$

of degree $n\ge 1$ by $P_n$; it is a linear space isomorphic to $C^{n+1}$. It is natural to deal with the bounded sets of polynomials, so one can consider without loss of generality the unit ball formed by $p_n$ with $|p_n\left(z\right)|<1$ on the unit disk $\mathbb{D}=\left\{\right|z|<1\}$.

We regard the polynomials $p_n$ also as the Schwarzian derivatives

$$S_w\left(z\right)={\left({\displaystyle \frac{w^{″}\left(z\right)}{w^{\prime }\left(z\right)}}\right)}^{\prime }-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{w^{″}\left(z\right)}{w^{\prime }\left(z\right)}}\right)}^2$$

of locally univalent holomorphic functions $w\left(z\right)=a_0+a_{1}z+\dots$ on $\mathbb{D}$ arising as the solutions of the differential equation

$${(w^{″}\left(z\right)/w^{\prime }\left(z\right))}^{\prime }-{(w^{″}\left(z\right)/w^{\prime }\left(z\right))}^2/2=p_n\left(z\right)$$

(1)

satisfying the prescribed initial conditions $w\left(0\right)=a_0,w^{\prime }\left(0\right)=a_1,w^{″}\left(0\right)=a_2$ (equivalently, as ratios of two linearly independent solutions ${\eta }_1,{\eta }_2$ of the linear differential equation $2{\eta }^{″}+f\left(z\right)\eta =0$ satisfying ${\eta }_1\left(0\right)=0,{\eta }_1^{\prime }\left(0\right)=1$ and ${\eta }_2\left(0\right)=1,{\eta }_2^{\prime }\left(0\right)=0$).

From this point of view, one can consider a more general situation, taking the rotationally invariant bounded convex subdomains $G\subset P_n$, which contains with every their polynomial $p_n$ its pre- and post-rotations

$$p_n\left(z\right)\mapsto p_{n,\alpha ,\beta }\left(z\right)=e^{i\beta }{p}_n\left(e^{i\alpha }z\right)$$

(2)

with independent $\alpha ,\beta \in [-\pi ,\pi ]$. Noting that the Minkowski functional of any such domain G generates a norm on ${\mathbb{C}}^{n+1}$ in which G is the unit ball, and that all Banach norms on $P_n$ are equivalent, we deal with rotationally invariant bounded convex domains $G\subset P_n$ with prescribed Banach norms ${\parallel ·\parallel }_X$.

The Schwarzian derivatives are hyperbolically bounded holomorphic functions in $\mathbb{D}$, which means they have the growth

$$f\left(z\right)={O(1/(1-\left|z\right|)}^2\left)\mathrm{as}\right|z|\to 1.$$

The complex Banach space $\mathbf{B}$ of all such functions with norm ${\parallel f\parallel }_{\mathbf{B}}={sup}_{\mathbb{D}}{(1-|z|}^2{)}^2\left|f\left(z\right)\right|$ is dual to the space $A_1\left(\mathbb{D}\right)$ of integrable holomorphic functions on $\mathbb{D}$ (and $A_1\left(\mathbb{D}\right)\subset \mathbf{B}$).

Any $f\in \mathbf{B}$ is the Schwarzian derivative of a locally univalent function $w\left(z\right)$ on $\mathbb{D}$ determined by the differential equation $S_w=f$. An important basic fact here is that the Schwarzians $S_w$ of univalent solutions $w\left(z\right)$ with quasiconformal extension to the complementary disk ${\mathbb{D}}^{*}=\{z\in \widehat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}:\left|z\right|>1\}$ fill in the space $\mathbf{B}$ a bounded domain (containing the origin), which is regarded as a canonical model of the universal Teichmüller space $\mathbf{T}$, This allows one to apply the analytic and geometric features of Teichmüller spaces to solving the problems of complex analysis.

One can distinguish in any bounded rotationally invariant convex domain $G\subset P_n$ two extremal polynomials determined up to rotations (2):

$$p_n^0\left(z\right)=c_0^0+c_1^{0}z+\cdots +c_n^0{z}^n,$$

(3)

which has the maximal first coefficient on G, i.e., with $|c_1^0|={max}_G\left|c_1\left(p_n\right)\right|$, and the Schwarzian

$$S_{w_0}\left(z\right)=c_{0,0}+c_{m_1,0}{z}^{m_1}+\cdots +c_{m_j,0}{z}^{m_j}+\cdots +c_{n,0}{z}^n$$

(4)

of the univalent solution of Equation (1) with the maximal second coefficient

$$a_2\left(w_0\right)=a_2\left(G\right):=sup\left\{|a_2\left(w\right)|:S_w\in G\right\}.$$

There are domains, for which these extremal polynomials are different. For example, equipping $P_n$ with norm

$$\parallel p_n\parallel ={\left({\displaystyle \sum _0^n}{\left|c_j\right|}^2\right)}^{1/2},$$

one obtains the extremal value of $c_m,m\le n$, which is attained on the monomial $p_n\left(z\right)=c{z}^m$ with an appropriate c; here $c_{1,0}=0$.

An interesting set in Banach spaces is the subset ${\mathcal{Z}}_X$ in X formed by functions f, which are zero-free on the disk $\mathbb{D}$. Such functions arise and play an essential role in many fields of complex analysis, in particular, as the derivatives of locally univalent functions and of covering maps, and have been investigated by many authors.

The problem of estimating the coefficients of nonvanishing holomorphic functions in the general Banach spaces was stated by Hummel, Scheinberg and Zalcman in [11]. It is a special case of the evaluation problem for coefficients of holomorphic functions on $\mathbb{D}$.

We are concerned here with solving these problems for spaces formed by polynomials and apply the technique developed in [12,13]. It involves lifting the polynomial functionals onto the universal Teichmüller space $\mathbf{T}$ and to the Bers fiber space $Fib\left(\mathbf{T}\right)$ over $\mathbf{T}$ and applying the quasiconformal deformations of holomorphic functions.

The needed results from Teichmüller space theory are presented, for example, in [14,15,16,17].

2. Evaluation of Polynomial Coefficients

Recently, the author presented in [18] a new approach to the evaluation of polynomial coefficients based on applying the deep features of univalent functions and Teichmüller spaces. It yields sharp bounds for polynomial coefficients by the nonzero coefficients of two naturally arising extremal polynomials.

In particular, this approach provides the following theorem on evaluation of polynomial coefficients.

Theorem 1 

([18]). (a) If a Banach submanifold $X\in \mathbf{B}$ is such that the Schwarzian

$$S_{w_0}\left(z\right)=c_0+c_{1,0}z+\dots$$

of the extremal univalent solution $w_0$ of the corresponding Equation (2) maximizing $a_2$ has the first coefficient $c_{1,0}\ne 0$, then the function $f_0\left(z\right)$ maximizing the first coefficient $c_1$ on X coincides with this $S_{w_0}$, and all larger coefficients $c_n,n>1$, of every function $f\in X$ are estimated by coefficients $c_1^0,c_n^0$ of $S_{w_0}$ via

$$|c_n|\le max(|c_1^0|,|c_n^0\left|\right),$$

(5)

provided that $c_n^0\ne 0$.

(b) For any rotationally invariant bounded convex subdomain G of $P_n$ and all polynomials $p_n\in G$, whose zeros are located outside of the disk $\mathbb{D}$, their coefficients are sharply estimated similarly to (5) with bounds given by a nonvanishing in $\mathbb{D}$ polynomial $p_n^0$, on which the maximum of $|c_1|$ on $G_{\mathcal{Z}}$ is attained.

The part $\left(b\right)$ relates to the Hummel–Scheinberg–Zalcman problem mentioned above.

The proof of this theorem involves the extension of the coefficient functionals onto the space $\mathbf{T}$ and its cover $Fib\left(\mathbf{T}\right)$ and applying the Bers isomorphism theorem [19].

The present paper continues this research and aims to establish a complete explicit solution of this problem. We prove the following theorem, which essentially strengthens Theorem 1 and implies the sharp explicit bounds for coefficients of all polynomials $p_n$ with $n\ge 5$.

Theorem 2. 

For any $n\ge 5$ and any rotationally invariant domain, $G\subset P_n$, equipped with a norm ${\parallel ·\parallel }_X$, we have:

(a) $c_{1,0}\ne 0$.

(b) The extremal polynomials (3) and (4) for any such n are equal (up to rotations (2));

(c) The coefficients $c_2,\dots ,c_{n-1}$ of all polynomials $p_n\in G$ are sharply estimated by

$$|c_m|\le max(|c_{1,0}|,|c_{m,0}\left|\right)=max\left(\right|c_1^0|,|c_m^0\left|\right)(m=2,\dots ,n-1).$$

(6)

(d) The similar assertions and estimates are valid also for nonvanishing polynomials on $\mathbb{D}$ (with the upper bounds corresponding to ${\mathcal{Z}}_X$).

More precisely, one can use the norms ${\parallel ·\left|\right|}_X$ of $L_q$-type with $2\le q\le \infty$ (which includes the $L_{\infty }$ case and estimates the polynomials in the Euclidean norm on $P_n$) and the arbitrary Hilbert norms (regarding $P_n$ as the subspace of a Hilbert space X) for which the Proposition 1 on quasiconformal deformations is valid.

Theorem 2 shows that actually the evaluation of coefficients of arbitrary polynomials $p_n\in G,n\ge 5$, is reduced to the maximization problem: find the polynomial $p_n^0$ with the maximal value of $|p_n^{\prime }\left(0\right)|$ on G.

Another way is to maximize the coefficient $a_2$ among univalent solutions of the Schwarz Equation (1). Note that the equivalent linear differential equations

$$2{\eta }^{″}\left(z\right)+p_n\left(z\right)\eta \left(z\right)=0$$

closely relate to special functions, and one has to use the ratios of some combinations of such functions (cf. [20]).

3. Underlying Results

3.1. A General Remark

We shall use two canonical classes of univalent functions in the disk, which play a crucial role in geometric function theory. The class S consists of univalent functions

$$w\left(z\right)=z+a_2{z}^2+\dots z_m{z}^m+\dots$$

on the unit disk. We use the univalent solutions of Equation (1) with such normalization. Their inversions

$$W_f\left(z\right)=1/w(1/z)=b_0+b_1{z}^{-1}+\cdots +b_n{z}^{-n}+\dots ,\left|z\right|>1,$$

are nonvanishing (zero-free) and univalent on the complementary disk ${\mathbb{D}}^{*}=\{z\in \widehat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}:\left|z\right|>1\}$.

The class of all $\widehat{\mathbb{C}}$-holomorphic univalent functions on ${\mathbb{D}}^{*}$ with a simple pole at infinity is denoted by $\mathrm{\Sigma }$.

The functions admitting quasiconformal extension to the whole Riemann sphere $\widehat{\mathbb{C}}$ are dense in these classes with respect to the spherical metric on $\widehat{\mathbb{C}}$.

Theorem 2 essentially relies on the following results.

3.2. Quasiconformal Deformations Associated with Polynomials

We first describe the needed quasiconformal deformations of holomorphic functions, which provide a fruitful tool in geometric complex analysis.

Such deformations imply the existence of quasiconformal maps h, which are conformal outside of a given domain $D\subset \widehat{\mathbb{C}}$, have the prescribed values $h^{\left({\alpha }_j\right)}$ of a finite collection of their derivatives in the prescribed points of $\overline{D}$ and preserve the norm of the original holomorphic function or, more generally, disturb this norm in the controlled bounds. The origins of such results go back to quasiconformal deformations of the punctured Riemann surfaces of finite genus, that is, with the finitely generated fundamental groups, introduced in [21].

In the case of polynomials, their deformations imply the biholomorphic maps of small neighborhoods of polynomials with a prescribed distortion of norms. For simplicity, we shall use the norms of $L_p$ type, for which we have the following result obtained in [13].

Fix a natural $n>1$, and consider the vectors $\mathbf{d}=(d_0,d_1,\dots ,d_n)\in {\mathbb{C}}^{n+1}$ with the Euclidean norm $\left|\mathbf{d}\right|$.

Proposition 1 

([13]). For any polynomial $p_n\left(z\right)=c_0+c_{1}z+\cdots +c_n{z}^n\in P_n$ with $c_n\ne 0$, there exists a number ${\epsilon }_0>0$, such that for every point

$$\mathbf{d}=(d_1,\dots ,,d_{n-1})\in {\mathbb{C}}^{n-1}$$

and every $a\in \mathbb{R}$ with

$$\left|\mathbf{d}\right|\le \epsilon ,\left|a\right|\le \epsilon ,\epsilon <{\epsilon }_0,$$

there exists a quasiconformal automorphism, h, of the extended complex plane $\widehat{\mathbb{C}},h\left(0\right)=0$, which is conformal outside of some disk $\mathbb{D}(w_0,r)\subset p_n\left({\mathbb{D}}^{*}\right)$ and determines a polynomial $p_n^{*}\left(z\right)={\displaystyle \sum _0^n}{c}_m^{*}{z}^m$ with norm $\parallel p_n^{*}{\parallel }_X={\parallel p_n\parallel }_X+a$ and coefficients

$$c_m^{*}=h^{\left(m\right)}\left(p_m\left(0\right)\right)=m!(c_m+d_m),m=1,\dots ,n-1.$$

The Beltrami coefficient ${\mu }_h$ of h satisfies $\parallel {\mu }_h{\parallel }_{\infty }\le M\epsilon$. The quantities ${\epsilon }_0$ and M depend only on f and n.

This proposition implies that the points $(h^{\mu }\left(c_1\right),\dots ,h^{\mu }\left(c_{n-1}\right))$ fill a whole neighborhood of the original point $(c_1,\dots ,c_{n-1})$ in ${\mathbb{C}}^{n-1}$, while the deformation of the last coordinate $c_n$ is used for preserving the Banach norm.

3.3. Covering Result

The following assertion is a generalization of Koebe’s one-quarter theorem.

Proposition 2 

([13,22]). Let $w\left(z\right)$ be a univalent solution of Equation (5) satisfying $w\left(0\right)=0,w^{\prime }\left(0\right)=e^{i\theta }$ with the fixed $\theta \in [-\pi ,\pi ]$. and let $w_0\left(z\right)=e^{i\theta }z+a_2^0{z}^2+\dots$ be one of the maximizing functions for $a_2$. Then, the image domain $w\left(\mathbb{D}\right)$ covers entirely the disk $D_{1/\left(2\right|a_2\left(G\right)\left|\right)}=\left\{\right|w|<1/\left(2\right|a_2\left(G\right)\left|\right)\}$. The radius value $1/\left(2\right|a_2\left(G\right)\left|\right)$ is sharp for this collection of functions, and the circle $\left\{\right|w|=1/\left(2\right|a_2\left(G\right)\left|\right)$ contains the points not belonging to $w\left(\mathbb{D}\right)$ if and only if $|a_2|=|a_2\left(G\right)|$ (i.e., when w is one of the maximizing functions for $a_2$ on the set G).

The inverted functions

$$W\left(z\right)=1/w(1/z)=e^{-i\theta }z-a_2^0+b_1{z}^{-1}+b_2{z}^{-2}+\dots$$

with $\zeta \in {\mathbb{D}}^{*}$ map the disk ${\mathbb{D}}^{*}$ onto a domain whose boundary is entirely contained in the disk $\left\{\right|W+a_2\left(G\right)|\le |a_2\left(G\right)\left|\right\}$.

3.4. Estimating General Coefficient Functionals

One of the basic results obtained by the approach of [12,13] indicated above is given by the following

Proposition 3 

([12,13]). Any rotationally invariant polynomial functional

$$J\left(w\right)=J(a_2,\dots ,a_s)$$

on S whose zero set $Z_J=\{w\in S:J\left(w\right)=0\}$ is separated from the rotation set of extremal function of Proposition 2, is maximized on the space X only by the rotations of this function.

In other words, any extremal function of $J\left(w\right)$ must be simultaneously maximal for the second coefficient $a_2$ on this class, unless $J\left(w_0\right)=0$.

4. Proof of Theorem 2

The main step in the proof is to establish that for all $n\ge 5$ the extremal Schwarzian $S_{w_0}$ has the first coefficient

$$c_{1,0}\ne 0.$$

(7)

Then the desired estimates (6) can be derived (for $n\ge 5$) also from Theorem 1.

Using the linearity of $c_m\left(p_n\right)$ on $P_n$, we distinguish the domain

$$G_{\rho }=\rho G$$

of radius $\rho >0$ chosen so that $G_{\rho }\subset \mathbf{T}$ and the boundaries of $\mathbf{T}$ have common points, noting that, by the Ahlfors–Weill theorem [23], any function (quadratic differential) $f\in \mathbf{B}$ with norm ${\parallel f\parallel }_{\mathbf{B}}=k<2$ is the Schwarzian derivative $S_w=f$ of a function w, which is univalent on the disk $\mathbb{D}$ and admits k-quasiconformal extension across the unit circle ${\mathbb{S}}^1$ to $\widehat{\mathbb{C}}$ with Beltrami coefficient canonically connected with f (this domain also can be regarded as a ball in the ${\parallel ·\parallel }_X$ on $P_n$).

The proof of the theorem will be accomplished in several steps. Each of these steps is of independent interest.

• Step 1: Algebraic set connected with $c_1$. To establish the inequality (7), we apply the well-known results on Grunsky coefficients. These coefficients are defined for functions $W\left(z\right)=z+b_0+b_1{z}^{-1}+\cdots \in \mathrm{\Sigma }$ by the expansion

$$log{\displaystyle \frac{W\left(z\right)-W\left(\zeta \right)}{z-\zeta }}=-{\displaystyle \sum _{m,n=1}^{\infty }}{\alpha }_{mn}{z}^{-m}{\zeta }^{-n},(z,\zeta )\in {\left({\mathbb{D}}^{*}\right)}^2,$$

(8)

and satisfy the inequality

$$\left|{\displaystyle \sum _{m,n=1}^{\infty }}\sqrt{mn}{\alpha }_{mn}{x}_m{x}_n\right|\le 1$$

(9)

for any sequence $\mathbf{x}=\left(x_n\right)$ from the unit sphere $S\left(l^2\right)$ of the Hilbert space $l^2$ with norm $\parallel \mathbf{x}\parallel ={\left({\displaystyle \sum _1^{\infty }}{\left|x_n\right|}^2\right)}^{1/2}$; hence $|{\alpha }_{mn}|\le 1/\sqrt{mn}$ for all $m,n$. Here, the principal branch of the logarithmic function is chosen. By the Grunsky theorem [24] (extended to multiply connected domains in [24,25]) the inequality (9) is the necessary and sufficient condition for univalence of a holomorphic function on the unit disk.

The Grunsky coefficients of functions $w\left(z\right)\in S$ are defined in a similar fashion; in particular, the functions $w\left(z\right)\in S$ and $W\left(z\right)=1/w(1/z)\in \mathrm{\Sigma }$ have the same Grunsky coefficients. Letting

$$g(z,\zeta )=log\left[\right(w\left(z\right)-w\left(\zeta \right))/(z-\zeta \left)\right]$$

one obtains from (8) that near any finite $\zeta$,

$${\displaystyle \frac{{\partial }^{2}g(z,\zeta )}{\partial z\partial \zeta }}={\displaystyle \frac{1}{6}}{S}_w\left(z\right)+O\left({(z-\zeta )}^2\right),$$

and hence, the Schwarzian of f is given by

$$S_w\left(z\right)=6\underset{\zeta \to z}{lim}{\displaystyle \frac{{\partial }^{2}g(z,\zeta )}{\partial z\partial \zeta }}=6{\displaystyle \sum _{m+n=2}^{\infty }}{\alpha }_{mn}{z}^{m+n-2}=6{\alpha }_{1,1}+24{\alpha }_{2,1}z+\dots .$$

(10)

A simple calculation implies (cf. [26], Ch. 1)

$${\alpha }_{2,1}={\alpha }_{1,2}=a_4-3a_2{a}_3+2a_2^3.$$

(11)

Note also that the relations (11) and (12) provide that the coefficients $a_m$ of $w\left(z\right)\in S$ and the coefficients $b_j$ of $W\left(z\right)=1/w(1/z)\in \mathrm{\Sigma }$ are the polynomials from the Grunsky coefficients ${\alpha }_{mn}$.

Returning to the initial polynomials $p_n\in P_n\left(1\right)$, we have from (10) and (11),

$$b_1=24(a_4-3a_2{a}_3+2a_2^3).$$

Hence, the equality $b_1=0$ defines in $P_n$ the algebraic set whose preimage in S has the equation

$$a_4-3a_2{a}_3+2a_2^3=0.$$

(12)

This set contains, in particular, the functions $w_0\left(z\right)$ corresponding to the even polynomials $p_n\left(z\right)$.

Now, we take the solution $w_0\left(z\right)$ of Equation (1) defined by the extremal polynomial (5), i.e., with the maximal ${\alpha }_2|$ on G, and using Proposition 1, construct for any $a_m,1\le m\le n-1$, a quasiconformal deformation $h^{\mu }\left(\zeta \right)$ with Beltrami coefficient $\mu$ supported in some disk ${\mathbb{D}}_{{\zeta }_0}\left(\rho \right)=\left\{\right|z-{\zeta }_0|<\rho \}$ sufficiently far from the origin so that $h^{\mu }$ the corresponding values

$$\begin{array}{cc}\hfill |a_4(h^{\mu }\circ p_n)-2a_2(h^{\mu }\circ p_n)a_3(h^{\mu }\circ p_n)+2a_2{(h^{\mu }\circ p_n)}^3|& =O(\parallel \mu {\parallel }_{\infty })>0,\hfill \\ \hfill |c_m(h^{\mu }\circ p_n)|& =|c_m\left(p_n\right)+{\left|O\right(\parallel \mu \parallel }_{\infty })>|c_m\left(p_n\right)|,\hfill \end{array}$$

while $\parallel h^{\mu }\circ p_n{\parallel }_X={\parallel S_{w_0}\parallel }_X$ (for the extremal $p_n$). This contradicts the extremality of $|a_2^0|$ and of $|c_1^0|$, which proves (7), i.e., the assertion $\left(a\right)$.

• Step 2: Proof of assertions $\left(b\right)$ and $\left(c\right)$. Having the inequality (7), one can use Propositions 2 and 3, which imply the upper bounds in (6) for all nonzero coefficients $c_1^0,\dots ,c_{n-1}^0$ and sharpness of these bounds.

We provide a detailed proof, which involves lifting the coefficient functionals onto the Teichmüller spaces $\mathbf{T}$ and $Fib\left(\mathbf{T}\right)$. The needed results on these spaces are given, for example, in [16,17,19,27,28,29].

Consider the classes $S_{z_0,\theta }$ of univalent functions $w\left(z\right)$ on the unit disk $\mathbb{D}$ with expansions

$$w\left(z\right)=a_1{z}^{-1}+a_2{z}^2+\dots$$

with $a_1=e^{i\theta },-\pi <\theta \le \pi$, having the fix point at $z_0$ on the boundary unit circle ${\mathbb{S}}^1$ and admitting quasiconformal extensions across ${\mathbb{S}}^1$ to the whole Riemann sphere $\widehat{\mathbb{C}}$ and take their union

$$\widehat{S}=\bigcup _{z_0\in {\mathbb{S}}^1,\theta \in (-\pi ,\pi ]}{S}_{z_0,\theta }.$$

The Beltrami coefficients ${\mu }_f\left(z\right)={\partial }_{\overline{z}}f/{\partial }_{z}f$ of these extensions run over the unit ball

$$Belt{\left(D^{*}\right)}_1=\{\mu \in L_{\infty }\left(\mathbb{C}\right):\mu \left(z\right){|D=0,\parallel \mu \parallel }_{\infty }<1\}.$$

Now pass to the inverted functions $W_w\left(z\right)=1/w(1/z)$ which form the corresponding classes ${\mathrm{\Sigma }}_{z_0,\theta }$ of nonvanishing univalent functions on the disk ${\mathbb{D}}^{*}$ with expansions

$$W\left(z\right)=e^{-i\theta }z+b_0+b_1{z}^{-1}+b_2{z}^{-2}+\dots ,F\left(z_0\right)=z_0,$$

and let ${\mathrm{\Sigma }}^0={\bigcup }_{z_0,\theta }{\mathrm{\Sigma }}_{z_0,\theta }$.

The coefficients $a_n$ of $f\left(z\right)$ and the corresponding coefficients $b_j$ of $F_f\left(z\right)$ are related by

$$b_0+e^{2i\theta }{a}_2=0,b_n+{\displaystyle \sum _{j=1}^n}{ϵ}_{n,j}{b}_{n-j}{a}_{j+1}+{ϵ}_{n+2,0}{a}_{n+2}=0,n=1,2,\dots ,$$

where ${ϵ}_{n,j}$ are the entire powers of $e^{i\theta }$. This successively implies the representations of $a_n$ by $b_j$ via

$$a_n={(-1)}^{n-1}{ϵ}_{n-1,0}{b}_0^{n-1}-{(-1)}^{n-1}(n-2){ϵ}_{1,n-3}{b}_1{b}_0^{n-3}+\mathrm{lower}\mathrm{terms}\mathrm{with}\mathrm{respect}\mathrm{to}{b}_0.$$

(13)

This transforms any coefficient functional $J\left(w\right)$ on S depending on a finite set of distinguished coefficients $a_{m_1},\dots ,a_{m_s}$ into a coefficient functional $\tilde{J}\left(W^{\mu }\right)$ on ${\mathrm{\Sigma }}^0$ depending on the corresponding coefficients $b_j$ and lifts the functionals $\tilde{J}\left(W\right)$ and $J\left(w\right)$ holomorphically onto the universal Teichmüller space $\mathbf{T}$ modelled via bounded domain in the space $\mathbf{B}$ of Schwarzians $S_W^{\mu }$. Our functional $J\left(w\right)=c_n$ is expressed in terms of these coefficients.

By Bers’s isomorphism theorem [19], its Fiber space $Fib\left(\mathbf{T}\right)$ is canonically isomorphic to Teichmüller space ${\mathbf{T}}_1$ of the punctured disk ${\mathbb{D}}_{*}=\mathbb{D}\setminus \left\{0\right\}$.

So, to lift J onto the covering space ${\mathbf{T}}_1$, we again pass to functional $\widehat{J}\left(\mu \right)=\tilde{J}\left(W^{\mu }\right)$ lifting J onto the ball $Belt{\left(\mathbb{D}\right)}_1$ and apply the ${\mathbf{T}}_1$-equivalence, i.e., the quotient map

$${\varphi }_{{\mathbf{T}}_1}:Belt{\left(\mathbb{D}\right)}_1\to {\mathbf{T}}_1,\mu \to {\left[\mu \right]}_{{\mathbf{T}}_1}.$$

Thereby, the functional $\tilde{J}\left(W^{\mu }\right)$ is pushed down to a bounded holomorphic functional $\mathcal{J}$ on the space ${\mathbf{T}}_1$ with the same range domain.

By Bers’s isomorphism theorem, the points of ${\mathbf{T}}_1$ as the pairs $X_{W^{\mu }}=(S_{W^{\mu }},F^{\mu }\left(0\right))$, where $\mu \in Belt{\left(\mathbb{D}\right)}_1$ obey ${\mathbf{T}}_1$-equivalence, and using the relations (13), one comes to a holomorphic functional

$$\mathcal{J}\left(X_{W^{\mu }}\right)=\mathcal{J}(S_{W^{\mu }},t),t=W^{\mu }\left(0\right).$$

on ${\mathbf{T}}_1=Fib\left(\mathbf{T}\right)$ and have to investigate its restriction to the image in $Fib\left(\mathbf{T}\right)$ of the original submanifold $G_n$.

By Proposition 2, the boundary of any domain $W^{\mu }\left({\mathbb{D}}^{*}\right)$ for any $W^{\mu }\left(z\right)\in {\mathrm{\Sigma }}^0$ is located in the disk $\left\{\right|W-b_0|\le a_2\left(G\right)\}$, and the second coordinate t runs over some subdomain $D_{\theta }$ in the disk ${\mathbb{D}}_{2a_2\left(G\right)}=\left\{\right|t|<2a_2\left(G\right)\}$ containing the origin. led proof.

So, it remains to investigate the case when some of these coefficients $c_m^0=0$. Consider the polynomials

$${\tilde{p}}_n\left(z\right)=c_1{z}^m.$$

One simply verifies that the map $p_n\mapsto {\tilde{p}}_n$ is holomorphic in $\mathbf{B}$-norm on $P_n$, and hence, the polynomials ${\tilde{P}}_n$ can be regarded as holomorphic functions of $p_n$.

Finally, we have to establish the range domain of $W^{\mu }\left(0\right)$ for $S_{W^{\mu }}$ running over G and describe the boundary points of this domain.

The features of construction of the Bers fiber space $Fib\left(\mathbf{T}\right)$ imply that image of $G_{\rho }$ in the space $Fib\left(\mathbf{T}\right)$ also is a connected submanifold covering $G_{\rho }$.

We select a dense subsequence, $\{{\theta }_1,{\theta }_2,\dots \}\subset [-\pi ,\pi ]$, getting the corresponding coefficient functionals on the classes $S_{z_0,{\theta }_j}$ and ${\mathrm{\Sigma }}_{z_0,{\theta }_j}$, and consider the sequence of increasing products of the quotient spaces

$${\mathcal{T}}_m={\displaystyle \prod _{j=1}^m}{\widehat{\mathrm{\Sigma }}}_{{\theta }_j}/\sim ={\displaystyle \prod _{j=1}^m}\left\{(S_{W_{{\theta }_j}},W_{{\theta }_j}^{{\mu }_j}\left(0\right))\right\}\simeq {\mathbf{T}}_1\times \cdots \times {\mathbf{T}}_1,$$

(14)

where the equivalence relation ∼ again means ${\mathbf{T}}_1$-equivalence. The Beltrami coefficients ${\mu }_j\in Belt{\left(\mathbb{D}\right)}_1$ are chosen here independently. For any ${\mathbf{T}}_1$, presented in the right-hand side of (15), the corresponding values of $W_{{\theta }_j}^{{\mu }_j}\left(0\right)$ run over some domain $D_{{\alpha }_j}\subset \mathbb{C}$, and the corresponding collection $\beta =({\beta }_1,\dots ,{\beta }_s)$ of the Bers isomorphisms ${\beta }_j:\left\{(S_{W_{{\theta }_j}},W_{{\theta }_j}^{{\mu }_j}\left(0\right))\right\}\to Fib\left(\mathbf{T}\right)$ determines a holomorphic surjection of the space ${\mathcal{T}}_m$ onto the product of m spaces $Fib\left(\mathbf{T}\right)$. This provides the corresponding holomorphic maps

$$\mathbf{J}=(\mathcal{J}\left(S_{W_{{\theta }_1}}\right),\dots ,\mathcal{J}\left(S_{W_{{\theta }_m}}\right))$$

whose (polydisk) norm satisfies

$$\underset{G^m}{max}\parallel \mathbf{J}\parallel =\underset{j}{max}|J\left(S_{W_{{\theta }_1}}\right)|=\underset{G}{max}\left|c_{m_j}\right|.$$

Now, pick an extremal polynomial, $p_n^{0,m}\in G$, for

$$|\mathcal{J}(S_{W^{\mu }},t)|=|c_{m_j}^0|=\underset{G}{max}\left|c_{m_j}\left(p_n\right)\right|,$$

noting that, by assumption $c_{m_j}^0\ne 0$. Consider the corresponding Equation (5) for this $p_n^{0,m}$, and take its univalent solution $w_{0,m}\left(z\right)$.

With $w_{0,m}$ being applied to this, the above construction of quotient spaces (14), one obtains, in view of the circular symmetry of the limit space for (14), a maximal function,

$$u\left(t\right)=max|c_{m_j}\left(t\right)|$$

on the disk ${\mathbb{D}}_{\alpha }$ of some radius, $\alpha \le 2a_2\left(G\right)$; this function is circularly symmetric and positive on any circle $\left\{\right|t|=r\}$ and attains its maximal value on the boundary circle.

Since this construction involves the polynomials and univalent $w\left(z\right)$ with $a_2\left(2\right)$ arbitrarily close to $a_{2a_2\left(G\right)}$, we have that $\alpha =2a_2\left(G\right)$. So, for every $c_m$,

$$\underset{{\mathbb{D}}_{2a_2\left(G\right)}}{max}u\left(t\right)=\underset{\chi \left(G\right)}{sup}\mathcal{J}(S_{W^{\mu }},t)=\underset{G}{max}\left|c_{m_j}\right|=\underset{\left|t\right|=2a_2\left(G\right)}{max}u\left(t\right),$$

(15)

provided that $c_{m,0}\ne 0,m\ge 1$.

Take in (1) the third initial condition with $a_2=a_2\left(G\right)$. Then, the equalities (15) and the uniqueness of solution of the Cauchy problem for Equation (1), together with the rotational invariance, provide that every extremal polynomial $p_n^{0,m}$ must coincide with $S_{w_0}$ and, hence, is determined up to pre- and post-rotations (2).

This also holds for $m_j=1$. Therefore, we have one (up to the indicated rotations) extremal polynomial, $p_n^0$, for all $m<n$.

By Proposition 3, both functionals

$$J_m\left(w\right)=\underset{G}{max}|c_m|,{\tilde{J}}_m=\underset{G}{max}\left|c_1\right|$$

attain their maximal values on the boundary function $S_{w_0}$ and its rotations (2). This implies the assertions $\left(b\right)$ and $\left(c\right)$.

• Step 3: Proof of assertion $\left(d\right)$. Pass to polynomials $p_{n,r}\left(z\right)={\displaystyle \frac{1}{r}}{p}_n\left(rz\right)$ with $r<1$ close to 1 and consider their collection $G_{1/r}$. Each $p_{n,r}$ has a neighborhood $U(p_{n,r},ϵ\left(r\right))$ in $G_{1/r}$ filled by polynomials which are zero-free in the disk $\mathbb{D}$. The union

$$G_{\mathcal{Z},r}=\bigcup _{p_n\in \mathcal{Z}}U(p_{n,r},ϵ\left(r\right))$$

of such neighborhoods is a rotationally invariant domain. One can apply to this domain the above arguments using instead of $a_2\left(G\right)$ the maximal value of $|a_2\left(w\right)|$ on the image of the set $G_{\mathcal{Z}}$ in S, which provides the corresponding subharmonic functionals $|c_m\left(p_n\right)|$ on $G_{\mathcal{Z}}$. The corresponding extremal polynomials $p_{n,r}^0$ do not vanish in $\mathbb{D}$.

This provides for $p_{n,r}$ the bounds (6) on $G_{\mathcal{Z},r}$, depending on r, and going to the limit as $r\to 1$, one obtains $\left(d\right)$. This completes the proof of Theorem 2.

5. Additional Remark

The restriction $n\ge 5$ arose from Proposition 1 on quasiconformal deformations, which requires keeping the larger coefficient $c_n$ for the variation of the norm on $P_n$.

In fact, the relation between the coefficients $a_m$ and $c_n$ yields that one can also use somewhat smaller $n<5$, but the presence of the algebraic set (12) provides the bound $n>2$. The exact lower bound for admissible n will be presented in a separate paper.

In any case, all the results of the present paper are valid for all $n\ge 5$.