Schur parameters and Carath\'eodory class

TL;DR

This paper explores the relationship between Schur parameters and Carathéodory functions, providing a recursive formula to express coefficients and analyzing the mapping properties of these correspondences.

Contribution

It introduces a recursive formula linking Carathéodory function coefficients to Schur parameters and studies the mapping properties of these relationships.

Findings

Derived a recursive formula for Carathéodory coefficients in terms of Schur parameters.

Established the parametrization of Carathéodory functions using independent variables.

Analyzed the mapping properties of the correspondence between parameters and functions.

Abstract

The Schur (resp. Carath\'eodory) class consists of all the analytic functions $f$ on the unit disk with $|f|\le 1$ (resp. $\Re f>0$ and $f(0)=1$). The Schur parameters $\gamma_0,\gamma_1,\dots (|\gamma_j|\le 1)$ are known to parametrize the coefficients of functions in the Schur class. By employing a recursive formula for it, we describe the $n$-th coefficient of a Carath\'eodory function in terms of $n$ independent variables $\gamma_1,\dots, \gamma_n$ with $|\gamma_j|\le 1.$ The mapping properties of those correspondences are also studied.

Equations

$$\mathcal{S}$$

$$\mathcal{P}$$

$${\Delta }_n=\left|\begin{array}{ccccc}2& p_1& p_2& \cdots & p_n\\ \overline{p_1}& 2& p_1& \cdots & p_{n-1}\\ \overline{p_2}& \overline{p_1}& 2& \cdots & p_{n-2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \overline{p_n}& \overline{p_{n-1}}& \overline{p_{n-2}}& \cdots & 2\end{array}\right|\ge 0$$

$$2p_2$$

$$4p_3$$

$$L_a(z)=\frac{z-a}{1-\bar{a}z}$$

$$e^{i\theta }{L}_{a_1}(z)\cdots L_{a_d}(z)(\theta \in \mathbb{R},a_1,\dots ,a_d\in \mathbb{D})$$

$$\sigma f(z)=\frac{1}{z}\cdot \frac{f(z)-f(0)}{1-\overline{f(0)}f(z)}=\frac{L_{f(0)}(f(z))}{z}.$$

$$|\sigma f(z)|\le \underset{r\to 1^{-}}{\lim }\underset{|\zeta |\le r}{\max }|\sigma f(\zeta )|\le \underset{r\to 1^{-}}{\lim }\frac{1}{r}=1,$$

$$F_1({\gamma }_1)$$

$$F_n({\gamma }_1,\dots ,{\gamma }_n)$$

$$-{\bar{\gamma }}_1\sum _{k=2}^{n-1}{F}_{n-k}({\gamma }_2,\dots ,{\gamma }_{n-k+1})F_k({\gamma }_1,\dots ,{\gamma }_k),n\ge 2.$$

$$F_1$$

$$F_2$$

$$F_3$$

$$F_4$$

$$-(1-|{\gamma }_1{|}^2)(1-|{\gamma }_2{|}^2){\gamma }_3(2{\bar{\gamma }}_1{\gamma }_2+{\bar{\gamma }}_2{\gamma }_3)+(1-|{\gamma }_1{|}^2){\bar{\gamma }}_1^2{\gamma }_2^3,$$

$$F_5$$

$$-(1-|{\gamma }_1{|}^2)(1-|{\gamma }_2{|}^2)(1-|{\gamma }_3{|}^2){\gamma }_4(2{\bar{\gamma }}_1{\gamma }_2+2{\bar{\gamma }}_2{\gamma }_3+{\bar{\gamma }}_3{\gamma }_4)$$

$$+(1-|{\gamma }_1{|}^2)(1-|{\gamma }_2{|}^2){\gamma }_3(3{\bar{\gamma }}_1^2-{\bar{\gamma }}_1{\gamma }_3+3{\bar{\gamma }}_1|{\gamma }_2{|}^2{\gamma }_3+{\bar{\gamma }}_2^2{\gamma }_3^2)$$

$$-(1-|{\gamma }_1{|}^2){\bar{\gamma }}_1^3{\gamma }_2^4.$$

$$F_n({\gamma }_1,\dots ,{\gamma }_n)=(1-|{\gamma }_1{|}^2)\cdots (1-|{\gamma }_{n-1}{|}^2){\gamma }_n+G_n({\gamma }_1,\dots ,{\gamma }_{n-1}).$$

$$c_k=F_k({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_k),k=1,2,n-1.$$

$$c_k^{\prime }=F_k({\gamma }_2,{\gamma }_3,\dots ,{\gamma }_{k+1}),k=1,2,n-1.$$

$$\frac{f(z)-{\gamma }_1}{z}=f_1(z)\left(1-{\bar{\gamma }}_{1}f(z)\right).$$

$$c_n=(1-|{\gamma }_1{|}^2)c_{n-1}^{\prime }-{\bar{\gamma }}_1\sum _{k=2}^{n-1}{c}_{n-k}^{\prime }{c}_k.$$

$$c_n=F_{n+1}({\gamma }_0,{\gamma }_1,\dots ,{\gamma }_n),n=0,1,2,\dots ,$$

$$g(z)=\frac{1+\omega (z)}{1-\omega (z)},z\in \mathbb{D}.$$

$$g(z)=1+2\sum _{n=1}^{\infty }{b}_n{z}^n\text{and}\omega (z)=\sum _{n=1}^{\infty }{c}_n{z}^n.$$

$$Q_n(x_1,\dots ,x_n)=x_n+\sum _{k=1}^{n-1}{x}_{n-k}{Q}_k(x_1,\dots ,x_k),n\ge 2.$$

$$b_n=\sum _{k=0}^{n-1}{c}_{n-k}{b}_k=c_n+\sum _{k=1}^{n-1}{c}_{n-k}{b}_k,$$

$$b_n=c_n+\sum _{k=1}^{n-1}{c}_{n-k}{Q}_k(c_1,\dots ,c_k)=Q_n(c_1,\dots ,c_n),$$

$$b_1$$

$$b_2$$

$$b_3$$

$$b_4$$

$$b_5$$

$$c_n=b_n-\sum _{k=1}^{n-1}{c}_{n-k}{b}_k,=b_n-\sum _{k=1}^{n-1}{b}_{n-k}{c}_k.$$

Full text

1. Introduction

We denote by ${\operatorname{Hol}}({\mathbb{D}})$ the set of analytic ($=$ holomorphic) functions on the unit disk ${\mathbb{D}}=\{z:|z|<1\}$ in the complex plane ${\mathbb{C}}.$ In the present note, our main concern is about the subclasses

[TABLE]

These are called the Schur class and the Carathéodory class, respectively, and members of ${\mathscr{S}}$ and ${\mathscr{P}}$ are called Schur functions and Carathéodory functions, respectively. We also consider the subclass ${\mathscr{S}}_{0}=\{\omega\in{\mathscr{S}}:\omega(0)=0\}=\{zf(z):f\in{\mathscr{S}}\}$ of ${\mathscr{S}}.$ These classes play an important role in Geometric Function Theory. For example, a function $f\in{\operatorname{Hol}}({\mathbb{D}})$ with $f(0)=0,f^{\prime}(0)=1,$ is starlike (resp.  convex) if $zf^{\prime}(z)/f(z)$ (resp. $1+zf^{\prime\prime}(z)/f^{\prime}(z)$) belongs to the class ${\mathscr{P}}.$ Also, a function $f$ in ${\operatorname{Hol}}({\mathbb{D}})$ is said to be subordinate to another $g\in{\operatorname{Hol}}({\mathbb{D}})$ (we write $f\prec g$ for it) if $f=g\circ\omega$ for some $\omega\in{\mathscr{S}}_{0}.$ Therefore, detailed information about the Taylor coefficients of functions in ${\mathscr{S}}$ and ${\mathscr{P}}$ will lead to various useful estimates in Geometric Function Theory. As is well known, a function $f(z)=c_{0}+c_{1}z+c_{2}z^{2}+\cdots$ in ${\mathscr{S}}$ satisfies $|c_{n}|\leq 1$ for all $n\geq 0$ and the bound is sharp for each $n.$ For a function $g(z)=1+p_{1}z+p_{2}z^{2}+\cdots$ in ${\mathscr{P}},$ the sharp inequality $|p_{n}|\leq 2~{}(n=1,2,\dots)$ is known as the Carathéodory lemma. A complete characterization of the coefficients of ${\mathscr{S}}$ and ${\mathscr{P}}$ in terms of determinants are classically known. For instance, the coefficients of Carathéodory functions are described in the following theorem due to Carathéodory and Toeplitz (see [8]).

Theorem A**.**

Let $g(z)=1+p_{1}z+p_{2}z^{2}+\cdots$ be a formal power series with coefficients in ${\mathbb{C}}.$ Then, $g$ represents a Carathéodory function on ${\mathbb{D}}$ if and only if

[TABLE]

for all $n\geq 0.$

Therefore, the region of the coefficients $p_{1},\dots,p_{n}$ is described by the $n$ inequalities $\Delta_{1}\geq 0,\dots,\Delta_{n}\geq 0.$ For instance, the region of the first coefficient is $|p_{1}|\leq 2.$ However, it is not easy to use them in general. Parametric representations of the coefficients are often more useful. Libera and Złotkiewicz [5] derived the following parametrizations of possible values of $p_{2}$ and $p_{3}$ from Theorem A.

Theorem B**.**

Let $g(z)=1+p_{1}z+p_{2}z^{2}+\cdots$ be a Carathéodory function with $p_{1}\geq 0.$ Then there are numbers $x,y\in{\overline{\mathbb{D}}}$ such that

[TABLE]

Also, recently Kwon, Lecko and Sim [4] obtained a similar parametrization of $p_{4}.$ We remark that the assumption $p_{1}\geq 0$ is harmless because we can normalize any function $g\in{\mathscr{P}}$ so that $p_{1}\geq 0$ by considering a suitable rotation $g(e^{i\theta}z).$ In recent years, these results are used frequently to estimate Hankel determinants of functions in special classes of univalent functions. See, for instance, [3] and [9] and references therein. On the other hand, it is known that the coefficients of a function in the Schur class ${\mathscr{S}}$ are described by the Schur parameters (see [7, §1.3]). In Section 2, we will give a recursive method to compute $c_{n}$ in terms of the Schur parameters, as well as basic facts about the Schur class. Since the classes ${\mathscr{S}}_{0}$ and ${\mathscr{P}}$ are related by the Cayley transformation, we can develop a systematic approach to get concrete relations between the coefficients of these two classes in Section 3. Then in Section 4, we will parametrize $b_{n}=p_{n}/2$ as a function $T_{n}(\gamma_{1},\dots,\gamma_{n})$ of $n$ independent variables $\gamma_{1},\dots,\gamma_{n}$ in ${\overline{\mathbb{D}}}.$ The definition of $T_{n}$ given in Section 4 is convenient to observe basic properties of it but somewhat indirect. We also give a recursive formula to describe $T_{n}$ in Section 5.

2. Schur algorithm

Let $a\in{\mathbb{D}}.$ We first recall that the Möbius transformation

[TABLE]

keeps ${\mathbb{D}}$ invariant as a set and maps $a$ to [math] and [math] to $-a.$ In particular, $L_{a}\in{\mathscr{S}}.$ A function in ${\mathscr{S}}$ of the form

[TABLE]

is called a finite Blaschke product of degree $d$ and the set of all such functions will be denoted by ${\mathscr{B}}_{d}.$ Note that ${\mathscr{B}}_{0}$ consists of unimodular constants; namely, ${\mathscr{B}}_{0}={\mathbb{T}}=\partial{\mathbb{D}},$ and that ${\mathscr{B}}_{1}$ is nothing but the group of analytic automorphisms of ${\mathbb{D}}.$ For a function $f\in{\mathscr{S}}\setminus{\mathscr{B}}_{0},$ we consider the new function $\sigma f$ defined by

[TABLE]

Since the origin is a removable singularity, the function $\sigma f$ is analytic on ${\mathbb{D}}.$ Moreover, the maximum modulus principle implies that

[TABLE]

and hence $\sigma f\in{\mathscr{S}}.$ We define $\sigma f=0$ when $f\in{\mathscr{B}}_{0}.$ In this way, a self-map $\sigma:{\mathscr{S}}\to{\mathscr{S}}$ is defined. For a given $f\in{\mathscr{S}},$ we start with $f_{0}=f$ and define $f_{n}$ inductively by $f_{n}=\sigma f_{n-1}$ for $n\geq 1.$ That is to say, $f_{n}=\sigma^{n}f.$ This procedure is usually called the Schur algorithm. We define a sequence $\gamma_{n}~{}(n=0,1,2,\dots)$ by setting $\gamma_{n}=f_{n}(0)$ and call those numbers Schur parameters. For convenience, sometimes we write $\vec{\gamma}=\vec{\gamma}(f)=(\gamma_{0},\gamma_{1},\dots)\in{\overline{\mathbb{D}}}^{{\mathbb{N}}_{0}},$ where ${\mathbb{N}}_{0}=\{0,1,2,\dots\},$ and call it the Schur vector of $f.$ By definition, we observe that $\vec{\gamma}(\sigma f)=(\gamma_{1},\gamma_{2},\dots)$ if $\vec{\gamma}(f)=(\gamma_{0},\gamma_{1},\dots).$ What is the same, the Schur vector of $\sigma f$ is the backward shift of the Schur vector of $f.$ Schur [6] proved that the original function $f$ can be reproduced by its Schur vector.

Theorem C**.**

For a function $f\in{\mathscr{S}},$ its Schur parameters $\gamma_{0},\gamma_{1},\dots$ satisfy one of the following two conditions:

1. (i)

$|\gamma_{n}|<1$ for all $n.$ 2. (ii)

$|\gamma_{0}|<1,\dots,|\gamma_{n-1}|<1,|\gamma_{n}|=1,\gamma_{n+1}=0,\gamma_{n+2}=0,\dots$ for some $n\geq 0.$

The latter occurs if and only if $f\in{\mathscr{B}}_{n}.$ Moreover, for any sequence $\vec{\gamma}=(\gamma_{0},\gamma_{1},\dots)$ satisfying one of the above conditions, there exists a unique function $f\in{\mathscr{S}}$ such that $\vec{\gamma}(f)=\vec{\gamma}.$

We remark that the subclass ${\mathscr{S}}_{0}$ is characterized by $\gamma_{0}=0$ with the Schur parameters.

We define a sequence of functions $F_{n}$ of the $n$ complex variables $\gamma_{1},\gamma_{2},\dots,\gamma_{n}~{}(n=1,2,3,\dots)$ inductively by

[TABLE]

By construction, we see that the function $F_{n}(\gamma_{1},\dots,\gamma_{n})$ is indeed a polynomial in $\gamma_{1},\bar{\gamma}_{1},\dots,$ $\gamma_{n-1},\bar{\gamma}_{n-1},\gamma_{n}$ with integer coefficients. We compute the first several as follows:

[TABLE]

As the reader easily guesses, the following, known as Schur’s recurrence relation (cf. [7]), is verified by a simple induction argument.

Lemma 2.1**.**

For each $n\geq 2,$ there exists a function $G_{n}(\gamma_{1},\dots,\gamma_{n-1})$ of $n-1$ complex variables $\gamma_{1},\dots,\gamma_{n-1}$ such that the following equality holds:

[TABLE]

The following result will be the basis of our arguments below. It is not new (see, for instance, (1.3.47) in [7]) but a proof is given for convenience of the reader.

Lemma 2.2**.**

Let $\omega(z)=c_{1}z+c_{2}z^{2}+\cdots$ be a function in ${\mathscr{S}}_{0}$ with its Schur vector $(0,\gamma_{1},\gamma_{2},\dots).$ Then $c_{n}=F_{n}(\gamma_{1},\gamma_{2},\dots,\gamma_{n})$ for $n=1,2,3,\dots.$

Proof. We show the assertion by induction on $n.$ When $n=1,$ the assertion is trivial because $c_{1}=\gamma_{1}.$ Suppose that the assertion is valid up to $n-1$ for some $n\geq 2.$ Then

[TABLE]

Let $f(z)=\omega(z)/z.$ Then $f\in{\mathscr{S}}$ and it has the Schur vector $(\gamma_{1},\gamma_{2},\dots).$ Put $\tilde{\omega}(z)=z(\sigma f)(z)=zf_{1}(z)=c_{1}^{\prime}z+c_{2}^{\prime}z^{2}+\cdots,$ whose Schur vector is $(0,\gamma_{2},\gamma_{3},\dots).$ Therefore, by assumption of the induction again,

[TABLE]

By construction, we have the relation

[TABLE]

We now substitute the Taylor expansions of $f(z)$ and $f_{1}(z)$ and compare the coefficient of $z^{n-2}$ for both sides to obtain the relation

[TABLE]

Using (2.2) and (2.3), we obtain the required formula so that the induction argument has been completed. ∎

We remark that for a general function $f(z)=c_{0}+c_{1}z+c_{2}z^{2}+\cdots$ in ${\mathscr{S}}$ with the Schur vector $(\gamma_{0},\gamma_{1},\dots),$ the relations

[TABLE]

follow from Lemma 2.2 because the function $zf(z)=c_{0}z+c_{1}z^{2}+\cdots$ belongs to ${\mathscr{S}}_{0}$ and has $(0,\gamma_{0},\gamma_{1},\dots)$ as its Schur vector.

3. Relationship between Schur and Carathéodory classes

As is well known, a function $g\in{\mathscr{P}}$ corresponds, in a one-to-one manner, to a function $\omega\in{\mathscr{S}}_{0}$ through the Cayley transformation:

[TABLE]

Since it is more natural to think about the quantity $p_{n}/2$ for Carathéodory functions, we expand $g(z)$ and $\omega(z)$ in the forms

[TABLE]

In order to describe $b_{n}$’s in terms of $c_{k}$’s, we define a sequence of polynomials $Q_{n}(x_{1},\dots,x_{n})~{}(n=1,2,3,\dots)$ in the variables $x_{1},\dots,x_{n}$ inductively by $Q_{1}(x_{1})=x_{1}$ and

[TABLE]

It is worth noting that $Q_{n}(x_{1},\dots,x_{n})$ is a polynomial in $x_{1},\dots,x_{n}$ with non-negative integer coefficients. Then we have the following result.

Lemma 3.1**.**

$b_{n}=Q_{n}(c_{1},\dots,c_{n})$* for $n=1,2,\dots.$*

Proof. By (3.1), we have the relation $(g(z)+1)\omega(z)=g(z)-1.$ We now substitute the above Taylor expansions of $g(z)$ and $\omega(z)$ to obtain

[TABLE]

where we set $b_{0}=1$ for convenience. We show the assertion by induction on $n.$ When $n=1$ the assertion is clear. Suppose the assertion is valid up to $n-1;$ that is, $b_{k}=Q_{k}(c_{1},\dots,c_{k})$ for $k=1,\dots,n-1.$ Then by the above relation and (3.2)

[TABLE]

which means that the assertion is valid for $n.$ Thus the proof is complete by the induction. ∎

For instance we have the formulae

[TABLE]

By (3.3), we have also

[TABLE]

Thus, if we define a sequence of polynomials $R_{n}(x_{1},\dots,x_{n})~{}(n=1,2,3,\dots)$ in the variables $x_{1},\dots,x_{n}$ inductively by $R_{1}(x_{1})=x_{1}$ and

[TABLE]

then we obtain the following formula in the same way.

Lemma 3.2**.**

$c_{n}=R_{n}(b_{1},\dots,b_{n})$* for $n=1,2,\dots.$*

We note that the polynomial mapping $\vec{Q}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$ It might be interesting to observe that the polynomials $Q_{n}$ and $R_{n}$ are related by a very simple relation.

Proposition 3.3**.**

$R_{n}(x_{1},\dots,x_{n})=-Q_{n}(-x_{1},\dots,-x_{n})$* for $n\geq 1.$*

Proof. We again use the induction on $n.$ When $n=1,$ the assertion is clear. Suppose that the assertion is valid up to $n-1.$ Then by definition and the induction assumption we see that

[TABLE]

which shows the assertion for $n.$ ∎

Therefore, we obtain the following formulae easily from the previous ones:

[TABLE]

We end the section with a simple observation. We define mappings $\vec{Q}_{n}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$ and $\vec{R}_{n}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$ by

[TABLE]

Then we get the following result by construction.

Lemma 3.4**.**

The mappings $\vec{Q}_{n}$ and $\vec{R}_{n}$ are both polynomial automorphisms of ${\mathbb{C}}^{n}$ and they are inverses to each other; namely, $\vec{Q}_{n}\circ\vec{R}_{n}=\vec{R}_{n}\circ\vec{Q}_{n}={\operatorname{id}}_{{\mathbb{C}}^{n}}.$

4. Main results

We now define a sequence of functions $T_{n}=T_{n}(\gamma_{1},\dots,\gamma_{n})$ of $n$ complex variables $\gamma_{1},\dots,\gamma_{n}$ by

[TABLE]

where $F_{k}(\gamma_{1},\dots,\gamma_{k})$ and $Q_{n}(x_{1},\dots,x_{n})$ are defined by (2.1) and (3.2) respectively. For instance,

[TABLE]

Note that the formulae for $T_{1},T_{2}$ and $T_{3}$ appear as (1.3.51-53) in [7]. To formulate our main result on coefficients, it is convenient to consider the coefficient body of order $n$ for a subclass ${\mathscr{F}}$ of ${\operatorname{Hol}}({\mathbb{D}}):$

[TABLE]

We remark that ${\mathbf{X}}_{n}({\mathscr{F}})$ is convex whenever ${\mathscr{F}}$ is a convex subset of ${\operatorname{Hol}}({\mathbb{D}}).$ We recall that a subset $A$ of ${\mathbb{R}}^{d}$ is called a convex body if $A$ is compact and convex and has non-empty interior. It is well known that a convex body in ${\mathbb{R}}^{d}$ is homeomorphic to the closed unit ball $\mathbb{B}^{d}=\{(x_{1},\cdots,x_{d})\in{\mathbb{R}}^{d}:x_{1}^{2}+\cdots+x_{d}^{2}\leq 1\}$ (see [1, §11.3]).

Theorem 4.1**.**

Let $n$ be a positive integer. The coefficient body ${\mathbf{X}}_{n}({\mathscr{P}})$ of order $n$ for the Carathéodory class ${\mathscr{P}}$ is expressed as $\{1\}\times 2{\mathbf{V}}_{n},$ where ${\mathbf{V}}_{n}$ is a convex body in ${\mathbb{C}}^{n}.$ Moreover,

[TABLE]

is a continuous mapping of ${\overline{\mathbb{D}}}^{n}$ onto ${\mathbf{V}}_{n}$ and satisfies $\vec{T}_{n}({\mathbb{D}}^{n})={\operatorname{Int}\,}{\mathbf{V}}_{n}$ and $\vec{T}_{n}(\partial{\overline{\mathbb{D}}}^{n})=\partial{\mathbf{V}}_{n}.$ In addition, $\vec{T}_{n}:{\mathbb{D}}^{n}\to{\operatorname{Int}\,}{\mathbf{V}}_{n}$ is a real analytic diffeomorphism but $\vec{T}_{n}$ is not injective on the boundary $\partial{\overline{\mathbb{D}}}^{n}$ of ${\overline{\mathbb{D}}}^{n}$ for $n=2,3,\dots.$

Proof. By the normalization condition for Carathéodory functions, we first observe that ${\mathbf{X}}_{n}({\mathscr{P}})$ can be expressed as the form $\{1\}\times 2{\mathbf{V}}_{n},$ where $V_{n}=\{(b_{1},\dots,b_{n})\in{\mathbb{C}}^{n}:(1,2b_{1},\dots,2b_{n})\in{\mathbf{X}}_{n}({\mathscr{P}})\}.$ Since ${\mathscr{P}}$ is a convex subset of ${\operatorname{Hol}}({\mathbb{D}}),$ it is evident that ${\mathbf{V}}_{n}$ is convex in ${\mathbb{C}}^{n}.$ Similarly, we can write ${\mathbf{X}}_{n}({\mathscr{S}}_{0})=\{0\}\times{\mathbf{U}}_{n}.$

As we saw in the previous section, the coefficients of a function $g(z)=1+2b_{1}z+2b_{2}z^{2}+\cdots$ in ${\mathscr{P}}$ and those of $\omega(z)=c_{1}z+c_{2}z^{2}+\cdots$ in ${\mathscr{S}}_{0}$ are related by $(b_{1},\dots,b_{n})=\vec{Q}_{n}(c_{1},\dots,c_{n})$ whenever $g$ and $\omega$ are related by $g=(1+\omega)/(1-\omega).$ In particular, we have the relation ${\mathbf{V}}_{n}=\vec{Q}_{n}({\mathbf{U}}_{n}).$ Let

[TABLE]

where $F_{1},\dots,F_{n}$ are defined in (2.1). Then $\vec{T}_{n}=\vec{Q}_{n}\circ\vec{F}_{n}$ by construction. Since $\vec{Q}_{n}:{\mathbb{C}}^{n}\to{\mathbb{C}}^{n}$ is a polynomial automorphism of ${\mathbb{C}}^{n}$ by Lemma 3.4, the other assertions follow from the next proposition, which may be of independent interest. ∎

Proposition 4.2**.**

Let $n$ be a positive integer. The coefficient body ${\mathbf{X}}_{n}({\mathscr{S}}_{0})$ of order $n$ for ${\mathscr{S}}_{0}$ is described by ${\mathbf{X}}_{n}({\mathscr{S}}_{0})=\{0\}\times{\mathbf{U}}_{n},$ where ${\mathbf{U}}_{n}$ is a convex body in ${\mathbb{C}}^{n}.$ Moreover, $\vec{F}_{n}$ maps ${\overline{\mathbb{D}}}^{n}$ continuously onto ${\mathbf{U}}_{n}$ and satisfies $\vec{F}_{n}({\mathbb{D}}^{n})={\operatorname{Int}\,}{\mathbf{U}}_{n}$ and $\vec{F}_{n}(\partial{\overline{\mathbb{D}}}^{n})=\partial{\mathbf{U}}_{n}.$ Furthermore, $\vec{F}_{n}:{\mathbb{D}}^{n}\to{\operatorname{Int}\,}{\mathbf{U}}_{n}$ is a real analytic diffeomorphism but $\vec{F}_{n}$ is not injective on $\partial{\overline{\mathbb{D}}}^{n}$ for $n=2,3,\dots.$

Before the proof, we make a preliminary observation.

Lemma 4.3**.**

Let $(\gamma_{1},\dots,\gamma_{n})\in{\mathbb{D}}^{n-1}\times{\mathbb{C}}$ and $(\gamma_{1}^{\prime},\dots,\gamma_{n}^{\prime})\in{\mathbb{C}}^{n}.$ If $\vec{F}_{n}(\gamma_{1},\dots,\gamma_{n})=\vec{F}_{n}(\gamma_{1}^{\prime},\dots,\gamma_{n}^{\prime}),$ then $(\gamma_{1},\dots,\gamma_{n})=(\gamma_{1}^{\prime},\dots,\gamma_{n}^{\prime}).$

Proof. We show $\gamma_{j}=\gamma_{j}^{\prime}$ by induction. Since $\gamma_{1}=F_{1}(\gamma_{1})=F_{1}(\gamma_{1}^{\prime})=\gamma_{1}^{\prime},$ the assertion is clear for $j=1.$ Assume next that the assertion holds true up to $j-1;$ that is $\gamma_{1}=\gamma_{1}^{\prime},\dots,\gamma_{j-1}=\gamma_{j-1}^{\prime}.$ Then, by Lemma 2.1, we observe

[TABLE]

Since $(1-|\gamma_{1}|^{2})\cdots(1-|\gamma_{j-1}|^{2})\neq 0$ by assumption, we conclude that $\gamma_{j}=\gamma_{j}^{\prime}.$ Thus the induction argument has been completed. ∎

We are ready to show the above proposition.

Proof of Proposition 4.2. Since ${\mathscr{S}}_{0}$ is compact and convex in the vector space ${\operatorname{Hol}}({\mathbb{D}})$ endowed with the topology of locally uniform convergence on compact subsets of ${\mathbb{D}},$ ${\mathbf{X}}_{n}({\mathscr{P}})$ is also compact and convex. Consequently, the set ${\mathbf{U}}_{n}$ is compact and convex in ${\mathbb{C}}^{n}.$ In order to see that ${\mathbf{U}}_{n}$ is a convex body, we have only to show that ${\mathbf{U}}_{n}$ has a non-empty interior. For instance, we see that the polynomial $\omega(z)=c_{1}z+c_{2}z^{2}+\cdots+c_{n}z^{n}$ with $|c_{1}|+|c_{2}|+\cdots+|c_{n}|<1$ is contained in the class ${\mathscr{S}}_{0}.$ Hence, the non-empty open set $|c_{1}|+|c_{2}|+\cdots+|c_{n}|<1$ is contained in ${\mathbf{V}}_{n}.$

We next show that $\vec{F}_{n}({\overline{\mathbb{D}}}^{n})={\mathbf{U}}_{n}.$ Indeed, for $(c_{1},\dots,c_{n})\in{\mathbf{U}}_{n},$ by definition, there is a function $\omega\in{\mathscr{S}}_{0}$ such that $\omega(z)=c_{1}z+\cdots+c_{n}z^{n}+O(z^{n+1}).$ Let $(0,\gamma_{1},\gamma_{2},\dots)$ be the Schur vector of $\omega.$ Then $(c_{1},\dots,c_{n})=\vec{F}_{n}(\gamma_{1},\dots,\gamma_{n}).$ Thus the inclusion relation $\vec{F}_{n}({\overline{\mathbb{D}}}^{n})\supset{\mathbf{U}}_{n}$ follows. Conversely, choose $n$ points $\gamma_{1},\dots,\gamma_{n}$ from ${\overline{\mathbb{D}}}$ arbitrarily. When $(\gamma_{1},\dots,\gamma_{n})\in{\mathbb{D}}^{n},$ let $\vec{\gamma}=(\gamma_{1},\dots,\gamma_{n},0,0,\dots).$ Otherwise, there is a unique $m$ such that $|\gamma_{1}|<1,\dots,|\gamma_{m-1}|<1$ and $|\gamma_{m}|=1.$ Then we set $\vec{\gamma}=(\gamma_{1},\dots,\gamma_{m},0,0,\dots).$ Theorem C now guarantees existence of a function $f$ such that $\vec{\gamma}(f)=\vec{\gamma}.$ We can now expand the function $\omega(z)=zf(z)$ in the form $\omega(z)=c_{1}z+c_{2}z^{2}+\cdots.$ Thus we see that $\vec{F}_{n}(\gamma_{1},\dots,\gamma_{n})=(c_{1},\dots,c_{n})\in{\mathbf{U}}_{n}.$ Hence, we have shown that the other inclusion relation $\vec{F}_{n}({\overline{\mathbb{D}}}^{n})\subset{\mathbf{U}}_{n}.$ We have shown $\vec{F}_{n}({\overline{\mathbb{D}}}^{n})={\mathbf{U}}_{n}.$

Note that $\vec{F}_{n}$ is real analytic on ${\mathbb{C}}^{n}.$ We show now that $\vec{F}_{n}:{\mathbb{D}}^{n}\to{\mathbb{C}}^{n}$ is locally diffeomorphic; in other words, the Jacobian $J_{\vec{F}_{n}}$ does not vanish on ${\mathbb{D}}^{n}.$ For a moment, we set $\delta_{j}=(1-|\gamma_{1}|^{2})\cdots(1-|\gamma_{j}|^{2})$ for short. Regarding $\vec{F}_{n}$ as a column vector, with the help of Lemma 2.1, we compute

[TABLE]

Thus we have shown that $\vec{F}_{n}$ is locally diffeomorphic on the domain ${\mathbb{D}}^{n}.$ In particular, the image $\vec{F}_{n}({\mathbb{D}}^{n})$ is contained in the interior ${\operatorname{Int}\,}{\mathbf{U}}_{n}$ of ${\mathbf{U}}_{n}.$ Moreover, Lemma 4.3 now implies that $\vec{F}_{n}$ is injective on ${\mathbb{D}}^{n}.$

We next prove that $\vec{F}_{n}$ maps $\partial{\overline{\mathbb{D}}}^{n}$ onto $\partial{\mathbf{U}}_{n}.$ Let $(\gamma_{1},\dots,\gamma_{n})\in\partial{\overline{\mathbb{D}}}^{n}.$ Then $|\gamma_{1}|<1,\dots,|\gamma_{j-1}|<1$ and $|\gamma_{j}|=1$ for some $1\leq j\leq n.$ To the contrary, we suppose that $p=\vec{F}_{n}(\gamma_{1},\dots,\gamma_{n})\in{\operatorname{Int}\,}{\mathbf{U}}_{n}.$ First consider the case when $j=n.$ Then for small enough $\delta>0,$ we have $p_{\delta}=\vec{F}_{n}(\gamma_{1},\dots,\gamma_{n-1},(1+\delta)\gamma_{n})\in{\operatorname{Int}\,}{\mathbf{U}}_{n}.$ Since $\vec{F}_{n}:{\overline{\mathbb{D}}}^{n}\to{\mathbf{U}}_{n}$ is surjective, $p_{\delta}=\vec{F}_{n}(\gamma_{1}^{\prime},\dots,\gamma_{n}^{\prime})$ for some $(\gamma_{1}^{\prime},\dots,\gamma_{n}^{\prime})\in{\overline{\mathbb{D}}}^{n}.$ We apply Lemma 4.3 again to deduce that $(\gamma_{1},\dots,\gamma_{n-1},(1+\delta)\gamma_{n})=(\gamma_{1}^{\prime},\dots,\gamma_{n}^{\prime}).$ In particular, $(1+\delta)\gamma_{n}=\gamma_{n}^{\prime}\in{\overline{\mathbb{D}}},$ which contradicts $(1+\delta)|\gamma_{n}|=1+\delta>1.$ Therefore, this case does not occur. When $1\leq j<n,$ we consider the projection $\pi:{\mathbb{C}}^{n}\to{\mathbb{C}}^{j}$ defined by $\pi(z_{1},\dots,z_{n})=(z_{1},\dots,z_{j}).$ By definition, we have $\pi({\mathbf{U}}_{n})={\mathbf{U}}_{j}.$ Since $\pi$ is an open mapping, we have $\pi(p)\in{\operatorname{Int}\,}{\mathbf{U}}_{j}.$ However, since $\pi(p)=\vec{F}_{j}(\gamma_{1},\dots,\gamma_{j}),$ this is again impossible by the same reason. Therefore $\vec{F}_{n}(\gamma_{1},\dots,\gamma_{n})\in\partial{\mathbf{U}}_{n}$ for $(\gamma_{1},\dots,\gamma_{n})\in\partial{\overline{\mathbb{D}}}^{n}$ at any event. Recalling that $\vec{F}_{n}:{\overline{\mathbb{D}}}^{n}\to{\mathbf{U}}_{n}$ is surjective, we now conclude that $\vec{F}_{n}({\mathbb{D}}^{n})={\operatorname{Int}\,}{\mathbf{U}}_{n}$ and that $\vec{F}_{n}(\partial{\overline{\mathbb{D}}}^{n})=\partial{\mathbf{U}}_{n}.$

Finally, we see that $\vec{F}_{n}$ is not injective on $\partial{\overline{\mathbb{D}}}^{n}$ when $n\geq 2.$ Indeed, by Lemma 2.1, $\vec{F}_{n}(\gamma_{1},\dots,\gamma_{n-1},\gamma_{n})=\vec{F}_{n}(\gamma_{1},\dots,\gamma_{n-1},0)$ for any $(\gamma_{1},\dots,\gamma_{n-1})\in\partial{\overline{\mathbb{D}}}^{n-1}$ and $\gamma_{n}\in{\overline{\mathbb{D}}}.$ ∎

We now make a comparison with the Libera-Złotkiewicz lemma (Theorem B above). In terms of $b_{n}=p_{n}/2,$ we can reformulate it as

[TABLE]

We observe that their results agree with our formulae for $T_{2}$ and $T_{3}$ when $b_{1}=\gamma_{1}\geq 0,x=\gamma_{2},y=\gamma_{3}.$

5. Recursion for $T_{n}(\gamma_{1},\dots,\gamma_{n})$

In the previous section, we defined $T_{n}(\gamma_{1},\dots,\gamma_{n})$ as the composition of the polynomial $Q_{n}(x_{1},\dots,x_{n})$ with $\vec{F}_{n}(\gamma_{1},\dots,\gamma_{n}).$ Recall that $Q_{n}$ and $F_{k}$ are both defined recursively. In principle, there should be a recursive formula which defines $T_{n}.$ We end this note by giving such a formula. Let $g(z)=1+2(b_{1}z+b_{2}z^{2}+\cdots)$ be a function in ${\mathscr{P}}$ and take $f\in{\mathscr{S}}$ with $\vec{\gamma}(f)=(\gamma_{1},\gamma_{2},\dots)$ so that $g(z)=(1+zf(z))/(1-zf(z)).$ Then, by construction of $T_{n},$ we obtain

[TABLE]

Let $g_{1}(z)=(1+zf_{1}(z))/(1-zf_{1}(z))=1+2(b_{1}^{\prime}z+b_{2}^{\prime}z^{2}+\cdots)$ for $f_{1}(z)=\sigma f(z)=(f(z)-\gamma_{1})/[z(1-\bar{\gamma}_{1}f(z))].$ Since $\vec{\gamma}(f_{1})=(\gamma_{2},\gamma_{3},\dots),$ we have

[TABLE]

In view of the relation $zf(z)=(g(z)-1)/(g(z)+1),$ we obtain

[TABLE]

from which we derive the formula

[TABLE]

Substituting the power series expansions of $g(z)$ and $g_{1}(z),$ we get the relation

[TABLE]

where we set $b_{0}=b_{0}^{\prime}=1.$ We look at the coefficients of $z^{n}$ of the functions in the both sides to obtain

[TABLE]

Hence,

[TABLE]

We now substitute (5.1) and (5.2) into the last formula to have the following result.

Theorem 5.1**.**

The functions $T_{n}(\gamma_{1},\dots,\gamma_{n})$ defined in (4.1) are described by the following recursive formula with the initial condition $T_{0}=1:$

[TABLE]

We remark that the transformation $g_{1}$ from $g$ above was already considered by Brown [2] in a more general context.