An application of the Schur algorithm to variability regions of certain analytic functions

TL;DR

This paper applies the Schur algorithm to characterize the variability regions of certain analytic functions mapping the unit disk into convex domains, providing explicit boundary descriptions and convexity properties.

Contribution

It introduces a novel application of the Schur algorithm to describe variability regions of functions into convex domains, including boundary parametrization.

Findings

Variability regions are convex closed Jordan domains.

Explicit boundary parametrization is provided.

The regions depend on the Schur algorithm and polynomial coefficients.

Abstract

Let $\Omega$ be a convex domain in the complex plane ${\mathbb C}$ with $\Omega \not= {\mathbb C}$, and $P$ be a conformal map of the unit disk ${\mathbb D}$ onto $\Omega$. Let ${\mathcal F}_\Omega$ be the class of analytic functions $g$ in ${\mathbb D}$ with $g({\mathbb D}) \subset \Omega$, and $H_1^\infty ({\mathbb D})$ be the closed unit ball of the Banach space $H^\infty ({\mathbb D})$ of bounded analytic functions $\omega$ in ${\mathbb D}$, with norm $\| \omega \|_\infty = \sup_{z \in {\mathbb D}} |\omega (z)|$. Let ${\mathcal C}(n) = \{ (c_0,c_1 , \ldots , c_n ) \in {\mathbb C}^{n+1}: \text{there exists} \; \omega \in H_1^\infty ({\mathbb D}) \; \text{satisfying} \; \omega (z) = c_0+c_1z + \cdots + c_n z^n + \cdots$ for ${z\in \mathbb D}\}$. For each fixed $z_0 \in {\mathbb D}$, $j=-1,0,1,2, \ldots$ and $c = (c_0, c_1 , \ldots , c_n) \in {\mathcal C}(n)$, we use the Schur algorithm to determine the region of variability $V_\Omega^j (z_0, c ) = \{ \int_0^{z_0} z^{j}(g(z)-g(0))\, d z : g \in {\mathcal F}_\Omega \; \text{with} \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$. We also show that for $z_0 \in {\mathbb D} \backslash \{ 0 \}$ and $c \in \textrm{Int} \, {\mathcal C}(n) $, $V_\Omega^j (z_0, c )$ is a convex closed Jordan domain, which we determine by giving a parametric representation of the boundary curve $\partial V_\Omega^j (z_0, c )$.

Equations

$${\mathcal{S}}^{\ast }=\{f\in \mathcal{S}:f(\mathbb{D})\text{mbox}isstarlikewithrespectto0\},$$

$$\mathcal{C}V=\{f\in \mathcal{S}:f(\mathbb{D})\text{mbox}isconvex\}.$$

$$M_1\le |f^{\prime }(z_0)|\le M_2,m_1\le \text{mbox}\mathrm{Arg}{f}^{\prime }(z_0)\le m_2\text{mbox}forallf\in \mathcal{F}$$

$$\{\log f^{\prime }(z_0):f\in \mathcal{C}V\}=\left\{\log \frac{1}{(1-z{)}^2}:|z|\le |z_0|\right\}.$$

$$\tilde{V}(z_0,\lambda )=\{\log f^{\prime }(z_0):f\in \mathcal{C}V\text{mbox}and{f}^{\prime \prime }(0)=2\lambda \}.$$

$$(-\pi ,\pi ]\ni \theta \mapsto$$

$$-\left(1-\frac{\lambda \cos (\theta /2)}{\sqrt{1-{\lambda }^2{\sin }^2(\theta /2)}}\right)\log \left\{1-\frac{e^{i\theta /2}{z}_0}{i\lambda \sin (\theta /2)-\sqrt{1-{\lambda }^2{\sin }^2(\theta /2)}}\right\}$$

$$-\left(1+\frac{\lambda \cos (\theta /2)}{\sqrt{1-{\lambda }^2{\sin }^2(\theta /2)}}\right)\log \left\{1-\frac{e^{i\theta /2}{z}_0}{i\lambda \sin (\theta /2)+\sqrt{1-{\lambda }^2{\sin }^2(\theta /2)}}\right\}.$$

$$\left\{{\int }_0^{z_0}\frac{g(\zeta )-g(0)}{\zeta }d\zeta :g\in \mathcal{A}(\mathbb{D})\text{mbox}withg(0)=1,g^{\prime }(0)=2\lambda ,g(\mathbb{D})\subset \mathbb{H}\right\}$$

$${\mathcal{F}}_{\Omega }(c)=\left\{g\in {\mathcal{F}}_{\Omega }:(P^{-1}\circ g)(z)=c_0+c_{1}z+\cdots +c_n{z}^n+\cdots ,z\in \mathbb{D}\right\}.$$

$$V_{\Omega }^j(z_0,c)=\left\{{\int }_0^{z_0}{\zeta }^j\left(g(\zeta )-g(0)\right)d\zeta :g\in {\mathcal{F}}_{\Omega }(c)\right\}.$$

$$\mathcal{C}(n):=$$

$$\text{mbox}and\omega (z)=c_0+c_{1}z+\cdots +c_n{z}^n+\cdots \in \mathbb{D}\},$$

$$\omega (z)=c_0+c_{1}z+\cdots +c_n{z}^n+\cdots ,z\in \mathbb{D}.$$

$${\sigma }_a(z)=\frac{z+a}{1+\overline{a}z},z\in \mathbb{D}.$$

$${\omega }_1(z)=\frac{\omega (z)-c_0}{z(1-\overline{c_0}\omega (z))}$$

$$\omega (z)={\sigma }_{c_0}(z{\omega }_1(z))=\frac{z{\omega }_1(z)+c_0}{1+\overline{c_0}z{\omega }_1(z)},$$

$${\omega }_1(z)=\frac{\omega (z)-c_0}{z(1-\overline{c_0}\omega (z))}$$

$$c_0^{(1)}=\frac{c_1}{1-|c_0{|}^2},c_p^{(1)}=\frac{c_{p+1}+\overline{c_0}{\sum }_{\ell =1}^p{c}_{p-\ell }^{(1)}{c}_{\ell }}{1-|c_0{|}^2}(1\le p\le n-1).$$

$$\omega (z)={\sigma }_{c_0}(z{\omega }_1(z))=\frac{z{\omega }_1(z)+c_0}{1+\overline{c_0}z{\omega }_1(z)}$$

$${\gamma }_p=\{\begin{array}{ll}\infty ,& \text{mbox}if{c}_p^{(0)}\ne 0\\ 0,& \text{mbox}if{c}_p^{(0)}=0.\end{array}$$

$${\gamma }_p=\{\begin{array}{ll}\infty ,& \text{mbox}if{c}_{p-j}^{(j)}\ne 0\\ 0,& \text{mbox}if{c}_{p-j}^{(j)}=0.\end{array}$$

$$c_0^{(j+1)}=\frac{c_1^{(j)}}{1-|{\gamma }_j{|}^2},c_p^{(j+1)}=\frac{c_{p+1}^{(j)}+\overline{{\gamma }_j}{\sum }_{\ell =1}^p{c}_{p-\ell }^{(j+1)}{c}_{\ell }^{(j)}}{1-|{\gamma }_j{|}^2}(1\le p\le n-j-1).$$

$$\omega (z)={\sigma }_{{\gamma }_0}(z{\sigma }_{{\gamma }_1}(\cdots z{\sigma }_{{\gamma }_n}(z{\omega }^{\ast }(z))\cdots )),$$

$$\omega (z)={\sigma }_{{\gamma }_0}(z{\sigma }_{{\gamma }_1}(\cdots z{\sigma }_{{\gamma }_{i-1}}({\gamma }_{i}z)\cdots )).$$

$$\left\{\begin{array}[]{l}\omega_{0}(z)=\omega(z)\\[3.0pt] \omega_{1}(z)=\displaystyle\frac{\omega_{0}(z)-\omega_{0}(0)}{z(1-\overline{\omega_{0}(0)}\omega_{0}(z))}\\[3.0pt] \vdots\\[3.0pt] \omega_{n}(z)=\displaystyle\frac{\omega_{n-1}(z)-\omega_{n-1}(0)}{z(1-\overline{\omega_{n-1}(0)}\omega_{n-1}(z))}\end{array}\right.$$

$${\gamma }_p={\omega }_p(0),{\omega }_p(z)=c_0^{(p)}+c_1^{(p)}z+\cdots +c_{n-p}^{(p)}{z}^{n-p}+\cdots$$

$$w_0={\int }_0^{z_0}{\zeta }^j\{P({\sigma }_{{\gamma }_0}(\zeta {\sigma }_{{\gamma }_1}(\cdots \zeta {\sigma }_{{\gamma }_{i-1}}({\gamma }_i\zeta )\cdots )))-P(c_0)\}d\zeta ,$$

$${\omega }_{\gamma ,\epsilon }(z)=$$

$$Q_{\gamma ,j}(z,\epsilon )=$$

$${\omega }_{\gamma ,\epsilon }(z)=c_0+c_{1}z+\cdots +c_n{z}^n+\cdots .$$

$$V_{\Omega }^j(z_0,c)=Q_{\gamma ,j}(z_0,\overline{\mathbb{D}}):=\{Q_{\gamma ,j}(z_0,\epsilon ):\epsilon \in \overline{\mathbb{D}}\}.$$

Full text

1. Introduction

Let ${\mathbb{C}}$ be the complex plane. For $c\in{\mathbb{C}}$ and $r>0$, let ${\mathbb{D}}(c,r)=\{z\in{\mathbb{C}}:|z-c|<r\}$, and $\overline{\mathbb{D}}(c,r)=\{z\in{\mathbb{C}}:|z-c|\leq r\}$. In particular, we denote the unit disk by ${\mathbb{D}}:={\mathbb{D}}(0,1)$. Let ${\mathcal{A}}({\mathbb{D}})$ be the class of analytic functions in the unit disk ${\mathbb{D}}$ with the topology of uniform convergence on every compact subset of ${\mathbb{D}}$. Let $H^{\infty}({\mathbb{D}})$ be the Banach space of analytic functions $f$ in ${\mathbb{D}}$ with norm $\|f\|_{\infty}=\sup_{z\in{\mathbb{D}}}|f(z)|$, and $H_{1}^{\infty}({\mathbb{D}})$ be the closed unit ball of $H^{\infty}({\mathbb{D}})$, i.e., $H_{1}^{\infty}({\mathbb{D}})=\{\omega\in H^{\infty}({\mathbb{D}}):\|\omega\|_{\infty}\leq 1\}$. Denote by $\mathcal{S}$, the class of univalent (i.e., one-to-one) functions $f$ in ${\mathcal{A}}({\mathbb{D}})$, normalized so that $f(0)=f^{\prime}(0)-1=0$. The subclasses $\mathcal{S}^{*}$ of starlike, and ${\mathcal{C}V}$ of convex functions are respectively defined by

[TABLE]

and

[TABLE]

Then ${\mathcal{C}V}\subset\mathcal{S}^{*}\subset\mathcal{S}$. For $f\in{\mathcal{A}}({\mathbb{D}})$ with $f(0)=f^{\prime}(0)-1=0$, $f\in\mathcal{S}^{*}$ if, and only if, ${\rm Re}\,(zf^{\prime}(z)/f(z))>0$ in ${\mathbb{D}}$. Similarly for $f\in{\mathcal{A}}({\mathbb{D}})$ with $f(0)=f^{\prime}(0)-1=0$, $f\in{\mathcal{C}V}$ if, and only if, ${\rm Re}\,(zf^{\prime\prime}(z)/f^{\prime}(z)+1)>0$ in ${\mathbb{D}}$. The basic properties of these classes of functions can be found for example in [2, 9].

Let ${\mathcal{F}}$ be a subclass of ${\mathcal{A}}({\mathbb{D}})$ and $z_{0}\in{\mathbb{D}}$. Then upper and lower estimates of the form

[TABLE]

are respectively called a distortion theorem, and a rotation theorem at $z_{0}$ for ${\mathcal{F}}$, where $M_{j}$ and $m_{j}$ ($j=1,2$) are some non-negative constants. Estimates such as these deal only with absolute values, or arguments of $f^{\prime}(z_{0})$. In order to study the complex value $f^{\prime}(z_{0})$ itself, it is necessary to consider the variability region of $f^{\prime}(z_{0})$ when $f$ ranges over the class ${\mathcal{F}}$, i.e., the set $\{f^{\prime}(z_{0}):f\in{\mathcal{F}}\}$. For example, it is known that

[TABLE]

For a proof, we refer to [2, Chapter 2, Exercise 10, 11 and 13]. For other examples see [7, 8] and the references therein.

Let ${\mathbb{H}}=\{w\in{\mathbb{C}}:\text{\rm Re}\,w>0\}$. For $f\in{\mathcal{C}V}$, the function $g$ given by $g(z)=1+zf^{\prime\prime}(z)/f^{\prime}(z)$ satisfies $g({\mathbb{D}})\subset{\mathbb{H}}$, i.e., $\text{\rm Re}\,g(z)>0$ in ${\mathbb{D}}$. Applying Schwarz’s lemma to $(g(z)-1)/(g(z)+1)$ we obtain $|f^{\prime\prime}(0)|\leq 2$ for $f\in{\mathcal{C}V}$. Gronwall [4], and independently Finkelstein [3] obtained the sharp lower and upper estimates for $|f^{\prime}(z_{0})|$, when $f\in{\mathcal{C}V}$ satisfies the additional condition $f^{\prime\prime}(0)=2\lambda$, where $z_{0}\in{\mathbb{D}}$ and $\lambda\in\overline{\mathbb{D}}$ are arbitrarily preassigned. Let

[TABLE]

It is easy to see that $\widetilde{V}(e^{-i\theta}z_{0},e^{i\theta}\lambda)=\widetilde{V}(z_{0},\lambda)$ for all $\theta\in{\mathbb{R}}$. If $|\lambda|=1$, then by Schwarz’s lemma, for $f\in{\mathcal{C}V}$ the condition $f^{\prime\prime}(0)=2\lambda$ forces $f(z)\equiv z/(1-\lambda z)$, and hence $\widetilde{V}(z_{0},\lambda)=\{\log 1/(1-\lambda z_{0})^{2}\}$. Thus it suffices to consider the case $0\leq\lambda<1$. In 2006, one of the present authors [15] obtained the following extension to Gronwall’s [4] result.

Theorem A**.**

For any $z_{0}\in{\mathbb{D}}\backslash\{0\}$ and $0\leq\lambda<1$, the set $\widetilde{V}(z_{0},\lambda)$ is a convex closed Jordan domain surrounded by the curve

[TABLE]

In order to prove Theorem A, Yanagihara [15] implicitly showed the following.

Theorem B**.**

For any $z_{0}\in{\mathbb{D}}\backslash\{0\}$ and $0\leq\lambda<1$, the variability region

[TABLE]

is the same convex closed Jordan domain as in Theorem A.

Note that putting $g(z)=1+zf^{\prime\prime}(z)/f^{\prime}(z)$, Theorem A is a direct consequence of Theorem B. For similar results, we refer to [10, 14].

The aim of this paper is to extend Theorem B, and to refine Theorem A. Throughout the paper, we will, unless otherwise stated, assume that $\Omega$ is a convex domain in ${\mathbb{C}}$ with $\Omega\not={\mathbb{C}}$, and $P$ is a conformal map of ${\mathbb{D}}$ onto $\Omega$.

Let ${\mathcal{F}}_{\Omega}$ be the class of analytic functions $g$ in ${\mathbb{D}}$ with $g({\mathbb{D}})\subset\Omega$. Then the map ${\mathcal{F}}_{\Omega}\ni g\mapsto\omega=P^{-1}\circ g\in H_{1}^{\infty}({\mathbb{D}})\backslash E$ is bijective, where $E=\{\omega\in H_{1}^{\infty}({\mathbb{D}}):\omega(z)\equiv a\;\text{with}\;|a|=1\}$. For $c=(c_{0},c_{1},\ldots,c_{n})\in{\mathbb{C}}^{n+1}$, let

[TABLE]

For $f\in{\mathcal{A}}({\mathbb{D}})$ with $f(0)=f^{\prime}(0)-1=0$, let $g(z)=1+zf^{\prime\prime}(z)/f^{\prime}(z)$. Then the coefficients of the Taylor series of $f$ up to order $n+1$ are uniquely determined by the coefficients of $\omega=P^{-1}\circ g$ up to order $n$, and vice versa. Thus in order to extend Theorem B, we consider the following problem.

Problem 1.1**.**

Let $n\in{\mathbb{N}}\cup\{0\}$. For $j=-1,0,1,\ldots$, $z_{0}\in{\mathbb{D}}\backslash\{0\}$ and $c=(c_{0},\ldots,c_{n})\in{\mathbb{C}}^{n+1}$, determine the variability region

[TABLE]

We note that the coefficient body ${\mathcal{C}}(n)$ defined by

[TABLE]

is a compact and convex subset of ${\mathbb{C}}^{n+1}$. We note that Schur [12, 13] characterized ${\mathcal{C}}(n)$ completely. We also refer to [5, Chapter I] and [1, Chapter 1], for a detailed treatment. For $c=(c_{0},\ldots,c_{n})\in{\mathbb{C}}^{n+1}$, one can calculate the corresponding Schur parameter $\gamma=(\gamma_{0},\ldots,\gamma_{k})$ of $c$, where $0\leq k\leq n$, and necessary and sufficient conditions for $c\in\textrm{Int}\,{\mathcal{C}}(n)$, $c\in\partial{\mathcal{C}}(n)$ and $c\not\in{\mathcal{C}}(n)$ can be described in terms of $\gamma$ (see Section 2).

The organization of this paper is as follows. In Section 2, we summarize the definitions and known facts concerning the Schur parameter $\gamma$. We then state our main theorem and its variant, which gives the solution to Problem 1.1. In Section 3, we state some lemmas on $p$-valent starlike and convex functions, and introduce the Schur polynomials associated with the Schur parameter $\gamma$. We then prove several lemmas on the Schur polynomials. In Section 4, we prove our main theorem, and its variant.

2. The Schur parameter and statement of results

For the sake of completeness we state the Carathéodry interpolation problem and its solution given by Schur [12, 13], which plays an important role in the proof of our main theorem.

Problem 2.1** (The Carathéodory interpolation problem).**

Let $n\in{\mathbb{N}}\cup\{0\}$. For $c=(c_{0},\ldots,c_{n})\in{\mathbb{C}}^{n+1}$, find necessary and sufficient conditions for the existence of $\omega\in H_{1}^{\infty}({\mathbb{D}})$ such that $\omega(z)$ has a series expansion of the form

[TABLE]

Furthermore, find an explicit description of all solutions.

We call $c=(c_{0},\ldots,c_{n})$ the Carathéodory data of length $n+1$. For $a\in{\mathbb{D}}$, define $\sigma_{a}\in\mbox{Aut}({\mathbb{D}})$ by

[TABLE]

We first solve Problem 2.1, when $n=0$ and $c=(c_{0})$.

It follows from the maximum modulus principle that Problem 2.1 has no solution if $|c_{0}|>1$, and has a unique solution $\omega(z)\equiv c_{0}$, if $|c_{0}|=1$. So suppose that $|c_{0}|<1$ and $\omega\in H_{1}^{\infty}({\mathbb{D}})$ satisfies $\omega(0)=c_{0}$. Then by the maximum modulus principle,

[TABLE]

belongs to $H_{1}^{\infty}({\mathbb{D}})$. This relation between $\omega$ and $\omega_{1}$ can be convertible. Consequently, the set of all solutions $\omega\in H_{1}^{\infty}({\mathbb{D}})$ is given by

[TABLE]

where $\omega_{1}\in H_{1}^{\infty}({\mathbb{D}})$ is arbitrary.

The general solution to Problem 2.1 can now be obtained from the above consideration and by recursively applying the following proposition, which was implicitly proved in [12, 13] and [5, Chapter 1].

Proposition 2.1**.**

Let $c=(c_{0},\ldots,c_{n})\in{\mathbb{C}}^{n+1}$ be a Carathéodory data of length $n+1$.

1. (1)

If $|c_{0}|>1$, or $|c_{0}|=1$ with $(c_{1},\ldots,c_{n})\not=(0,\ldots,0)$, then the Carathéodory problem with data $c$ has no solution. 2. (2)

If $|c_{0}|=1$ with $(c_{1},\ldots,c_{n})=(0,\ldots,0)$, then the Carathéodory problem data $c$ has the unique solution $\omega(z)\equiv c_{0}$. 3. (3)

Assume that $|c_{0}|<1$. If $\omega$ is a solution to the Carathéodory problem with data $c$, then

[TABLE]

is a solution to the Carathéodory problem with data $c^{(1)}$, where $c^{(1)}=(c^{(1)}_{0},\ldots,c^{(1)}_{n-1})$ is a data of length $n$ defined by

[TABLE]

Conversely, if we define $c^{(1)}=(c^{(1)}_{0},\ldots,c^{(1)}_{n-1})\in{\mathbb{C}}^{n}$ by (2.1), and $\omega_{1}$ is a solution to the Carathéodory problem with data $c^{(1)}$, then

[TABLE]

is a solution to the Carathéodory problem with data $c=(c_{0},\ldots,c_{n})$.

We note that $c=(c_{0},\ldots,c_{n})$ and $(c_{0},c^{(1)})=(c_{0},c^{(1)}_{0},\ldots,c^{(1)}_{n-1})$ are uniquely determined each other by (2.1).

For a given Carathéodory data $c=(c_{0},\ldots,c_{n})\in{\mathbb{C}}^{n+1}$, the Schur parameter $\gamma=(\gamma_{0},\ldots,\gamma_{k})$, $k=0,1,\ldots,n$ is defined as follows.

First, let $c^{(0)}=(c_{0}^{(0)},\ldots,c_{n}^{(0)})=(c_{0},\ldots,c_{n})$ and $\gamma_{0}=c_{0}^{(0)}$. If $|\gamma_{0}|>1$, we set $k=0$ and $\gamma=(\gamma_{0})$. If $|\gamma_{0}|=1$, we set $k=n$, and for $p=1,2,\ldots,n$,

[TABLE]

If $|\gamma_{0}|<1$, we define $c^{(1)}=(c_{0}^{(1)},\ldots,c_{n-1}^{(1)})\in{\mathbb{C}}^{n}$ by (2.1). Now assume that $\gamma_{0},\ldots,\gamma_{j-1}$ and $c^{(j)}=(c_{0}^{(j)},\ldots,c_{n-j}^{(j)})$ are already defined and satisfy $|\gamma_{0}|<1,\ldots,$ $|\gamma_{j-1}|<1$ for some $j$ with $1\leq j\leq n-1$. Then put $\gamma_{j}=c_{0}^{(j)}$. If $|\gamma_{j}|>1$, put $k=j$ and $\gamma=(\gamma_{0},\ldots,\gamma_{j})$. If $|\gamma_{j}|=1$, put $k=n$ and for $p=j+1,\ldots,n$,

[TABLE]

If $|\gamma_{j}|<1$, we define $c^{(j+1)}=(c_{0}^{(j+1)},\ldots,c_{n-j-1}^{(j+1)})\in{\mathbb{C}}^{n-j}$ as in Proposition 2.1, i.e.,

[TABLE]

Applying this procedure recursively, one obtains the Schur parameter $\gamma=(\gamma_{0},\ldots,\gamma_{k})$, $k=0,\ldots,n$ of $c=(c_{0},\ldots,c_{n})$.

When $|\gamma_{0}|<1,\ldots,|\gamma_{j}|<1$, the equations in (2.2) and $\gamma_{j}=c_{0}^{(j)}$ show that $c^{(j)}=(c_{0}^{(j)},c_{1}^{(j)},\ldots,c_{n-j}^{(j)})$ and $(\gamma_{j},c_{0}^{(j+1)},\ldots,c_{n-j-1}^{(j+1)})$ are uniquely determined by each other. Thus when $|\gamma_{0}|<1,\ldots,|\gamma_{n}|<1$, $c=(c_{0},\ldots,c_{n})=c^{(0)}$ and $\gamma=(\gamma_{0},\ldots,\gamma_{n})$ are also uniquely determined by each other. For an explicit representation of $\gamma$ in terms of $c$, we refer to the following results of Schur [12, 13].

Theorem C** (Schur [12, 13]).**

For $c=(c_{0},\ldots,c_{n})\in{\mathbb{C}}^{n+1}$, let $\gamma=(\gamma_{0},\ldots,\gamma_{k})$ be the Schur parameter of $c$.

1. (i)

If $k=n$ and $|\gamma_{0}|<1,\ldots,|\gamma_{n}|<1$, then all solutions to the Carathéodory problem with data $c$ are given by

[TABLE]

where $\omega^{*}\in H_{1}^{\infty}({\mathbb{D}})$ is arbitrary. 2. (ii)

If $k=n$ and $|\gamma_{0}|<1,\ldots,|\gamma_{i-1}|<1$, $|\gamma_{i}|=1$, $\gamma_{i+1}=\cdots=\gamma_{n}=0$ for some $i=0,\ldots,n$, then the Carathéodory problem with data $c$ has the unique solution

[TABLE] 3. (iii)

If the hypotheses of either (i) or (ii) does not hold, then there is no solution to the Carathéodory problem with data $c$.

Furthermore, if the hypothesis of (i) holds and

[TABLE]

then

[TABLE]

holds for $p=0,1,\ldots,n$.

For a detailed proof of Theorem C, we refer to [5, Chapter 1]. It is not difficult to see that for a given Carathéodory data $c$, the hypotheses of (i), (ii) and (iii) in Theorem C are respectively equivalent to $c\in\text{Int}\,{\mathcal{C}}(n)$, $c\in\partial{\mathcal{C}}(n)$ and $c\not\in{\mathcal{C}}(n)$.

Let $\Omega$ be a convex domain with $\Omega\not={\mathbb{C}}$, and $P$ be a conformal map of ${\mathbb{D}}$ onto $\Omega$. When $c\not\in{\mathcal{C}}(n)$ or $c\in\partial{\mathcal{C}}(n)$, Theorem C gives the following simple solution to Problem 1.1.

Theorem 2.1**.**

Let $c=(c_{0},\ldots,c_{n})\in{\mathbb{C}}^{n+1}$ be a Carathéodory data. If $c\not\in{\mathcal{C}}(n)$ i.e., the hypothesis of (iii) in Theorem C holds, then $V_{\Omega}^{j}(z_{0},c)=\emptyset$. If $c\in\partial{\mathcal{C}}(n)$ i.e., the hypothesis of (ii) in Theorem C holds, then $V_{\Omega}^{j}(z_{0},c)$ reduces to a set consisting of a single point $w_{0}$, where

[TABLE]

and $\gamma=(\gamma_{0},\ldots,\gamma_{i},0,\ldots,0)$ is the Schur parameter of $c$.

Now we introduce a family of functions, which are extremal for Problem 1.1 in the case $c\in\text{Int}\,{\mathcal{C}}(n)$.

For $\varepsilon\in\overline{\mathbb{D}}$ and Schur parameter $\gamma=(\gamma_{0},\ldots,\gamma_{n})$ of $c\in\text{Int}\,{\mathcal{C}}(n)$, let

[TABLE]

Then $\omega_{\gamma,\varepsilon}\in H_{1}^{\infty}({\mathbb{D}})$ is a solution to the Carathéodory problem with the data $c$, i.e.,

[TABLE]

In particular, we have $\omega_{\gamma,\varepsilon}(0)=c_{0}$. We note that for each fixed $\varepsilon\in\overline{\mathbb{D}}$, $\omega_{\gamma,\varepsilon}(z)$ and $Q_{\gamma,j}(z,\varepsilon)$ are analytic functions of $z\in{\mathbb{D}}$, and for each fixed $z\in{\mathbb{D}}$, $\omega_{\gamma,\varepsilon}(z)$ and $Q_{\gamma,j}(z,\varepsilon)$ are analytic functions of $\varepsilon\in\overline{\mathbb{D}}$. When $\varepsilon\in\partial{\mathbb{D}}$, $\omega_{\gamma,\varepsilon}(z)$ is a finite Blaschke product of $z$. Indeed, it follows from (2.4) that $\omega_{\gamma,\varepsilon}(z)$ is a rational function of $z$, which is analytic on $\overline{\mathbb{D}}$. If $|\varepsilon|=1$, then again by (2.4) it is easy to see that $\omega_{\gamma,\varepsilon}$ maps $\partial{\mathbb{D}}$ into $\partial{\mathbb{D}}$. Thus $\omega_{\gamma,\varepsilon}$ is a finite Blaschke product of $z$.

We next state the following, which is the main result of this paper.

Theorem 2.2**.**

Let $n\in{\mathbb{N}}\cup\{0\}$, $j\in\{-1,0,1,2,\ldots\}$, $c=(c_{0},\ldots,c_{n})\in\text{\rm Int}\,{\mathcal{C}}(n)$ and $\gamma=(\gamma_{0},\ldots,\gamma_{n})$ be the Schur parameter of $c$. Then for each fixed $z_{0}\in{\mathbb{D}}\backslash\{0\}$, $Q_{\gamma,j}(z_{0},\varepsilon)$ is a convex univalent function of $\varepsilon\in\overline{\mathbb{D}}$ and

[TABLE]

Furthermore

[TABLE]

for some $g\in{\mathcal{F}}_{\Omega}(c)$ and $\varepsilon\in\partial{\mathbb{D}}$ if, and only if, $g(z)\equiv P(\omega_{\gamma,\varepsilon}(z))$.

3. Preliminaries

First we note the following (which is elementary).

Lemma 3.1**.**

Let $n\in{\mathbb{N}}\cup\{0\}$ and $g_{0}$, $g_{1}\in{\mathcal{A}}({\mathbb{D}})$ satisfying $g_{1}(z)-g_{0}(z)={\mathcal{O}}(z^{n+1})$ as $z\rightarrow 0$. Then for any analytic function $\varphi$ defined in a neighborhood of $g_{0}(0)=g_{1}(0)$, $\varphi(g_{0}(z))-\varphi(g_{1}(z))={\mathcal{O}}(z^{n+1})$ as $z\rightarrow 0$.

We next give some simple lemmas concerning multivalent starlike and convex functions. For a positive integer $p$, we write $(\mathcal{S}^{*})^{p}:=\{f_{0}^{p}:f_{0}\in\mathcal{S}^{*}\}$.

Lemma 3.2**.**

Let $f\in{\mathcal{A}}({\mathbb{D}})$ with $f(z)=z^{p}+\cdots$ for ${z\in\mathbb{D}}$. Then $f\in(\mathcal{S}^{*})^{p}$, i.e., there exists $f_{0}\in\mathcal{S}^{*}$ with $f=f_{0}^{p}$ if, and only if,

[TABLE]

Proof.

When $p=1$, the lemma is well known (see [2, Theorem 2.10]). For general $p$, if there exists $f_{0}\in\mathcal{S}^{*}$ with $f=f_{0}^{p}$, then by the identity $zf^{\prime}(z)/f(z)=pzf_{0}^{\prime}(z)/f_{0}(z)$ we have $\text{\rm Re}\,\{zf^{\prime}(z)/f(z)\}>0$.

Inequality (3.1) ensures that $f$ has no zeros in ${\mathbb{D}}\backslash\{0\}$. Thus there exists $f_{0}\in{\mathcal{A}}({\mathbb{D}})$ with $f_{0}^{p}=f$ and $f_{0}^{\prime}(0)=1$. Again from the identity $zf^{\prime}(z)/f(z)=pzf_{0}^{\prime}(z)/f_{0}(z)$ it follows that $\text{\rm Re}\,\{zf_{0}^{\prime}(z)/f_{0}(z)\}>0$ in ${\mathbb{D}}$. Thus $f_{0}\in\mathcal{S}^{*}$. ∎

Lemma 3.3**.**

Let $f\in{\mathcal{A}}({\mathbb{D}})$ with $f(z)=z^{p}+\cdots$ for ${z\in\mathbb{D}}$. If $f$ satisfies

[TABLE]

then there exists $f_{0}\in\mathcal{S}^{*}$ with $f=f_{0}^{p}$.

Proof.

Let $q\in{\mathbb{N}}\cup\{0\}$, and $A$ and $B$ be analytic functions in ${\mathbb{D}}$ such that $A(z)=az^{q}+\cdots$ and $B(z)=bz^{q}+\cdots$ in ${\mathbb{D}}$ with $a,b\not=0$. Suppose that $a^{-1}A\in(\mathcal{S}^{*})^{q}$, or $b^{-1}B\in(\mathcal{S}^{*})^{q}$. Then Libera [6] showed that $\textrm{Re}\,\{A^{\prime}(z)/B^{\prime}(z)\}>0$ in ${\mathbb{D}}$ implies $\textrm{Re}\,\{A(z)/B(z)\}>0$ in ${\mathbb{D}}$. Now note that $1+zf^{\prime\prime}(z)/f^{\prime}(z)=(zf^{\prime}(z))^{\prime}/f^{\prime}(z)$. Let $g(z)=p^{-1}zf^{\prime}(z)$. Since $\text{\rm Re}\,(zg^{\prime}(z)/g(z))=\mbox{\rm Re}(1+zf^{\prime\prime}(z)/f^{\prime}(z))>0$, it follows from Lemma 3.2 that $g\in(\mathcal{S}^{*})^{p}$. Thus, with $A(z)=zf^{\prime}(z)$ and $B(z)=f(z)$ Libera’s result, shows that $\text{\rm Re}\,(zf^{\prime}(z)/f(z))>0$ for ${z\in\mathbb{D}}$, and so the lemma now follows from Lemma 3.2. ∎

A well known theorem due to Robertson [11] states that each $f\in{\mathcal{C}V}$ can be approximated by a sequence of conformal maps of ${\mathbb{D}}$ onto convex polygons, uniformly on any compact subset of ${\mathbb{D}}$ as follows.

Lemma 3.4**.**

[11]** For any function $f\in{\mathcal{C}V}$, $\eta>0$ and a compact subset $E$ of ${\mathbb{D}}$, there exists $h\in{\mathcal{A}}({\mathbb{D}})$ with $\sup_{z\in E}|f^{\prime}(z)-h(z)|<\eta$, such that $h(z)$ can be written in the form

[TABLE]

where $\eta_{i}\in\partial{\mathbb{D}}$, $0<\beta_{i}\leq 2$ $(i=1,\ldots,m)$, and $\sum_{i=1}^{m}\beta_{i}=2$.

The following brief summary of Schur polynomials will assist in the proof of our main theorem. For more details we refer to [1].

Let $c=(c_{0},\ldots,c_{n})\in\textrm{Int}\,{\mathcal{C}}(n)$ and $\gamma=(\gamma_{0},\ldots,\gamma_{n})$ be the Schur parameter of $c$. Note that $c=(c_{0},\ldots,c_{n})\in\textrm{Int}\,{\mathcal{C}}(n)$ forces $|\gamma_{0}|<1,\ldots,|\gamma_{n}|<1$. Suppose that $\omega\in H_{1}^{\infty}({\mathbb{D}})$ satisfies $\omega(z)=c_{0}+\ldots+c_{n}z^{n}+\cdots$, and define $\omega_{0},\omega_{1},\ldots,\omega_{n}\in H_{1}^{\infty}({\mathbb{D}})$ by (2.3) and $\omega_{n+1}\in H_{1}^{\infty}({\mathbb{D}})$ by

[TABLE]

Then since $\gamma_{k}=\omega_{k}(0)$, $k=0,\ldots,n$, we have

[TABLE]

We now define sequences of polynomials recursively by

[TABLE]

and

[TABLE]

where $k=0,\ldots,n-1$. Then from (3.2), we have

[TABLE]

The polynomials $A_{k}(z),B_{k}(z),\widetilde{A}_{k}(z)$ and $\widetilde{B}_{k}(z)$ ($k=0,1,\ldots,n$) are of degree at most $k$, and are called the Schur polynomials associated with $\gamma$. For convenience, we put $\omega^{*}(z)=\omega_{n+1}(z)$. Then by Theorem C and (3.5) we obtain the following.

Lemma 3.5**.**

Let $c=(c_{0},\ldots,c_{n})\in\text{\rm Int}\,{\mathcal{C}}(n)$, and $\gamma=(\gamma_{0},\ldots,\gamma_{n})$ be the Schur parameter of $c$. Then for any $\omega\in H^{\infty}_{1}({\mathbb{D}})$ with $\omega(z)=c_{0}+\cdots+c_{n}z^{n}+\cdots$ in ${\mathbb{D}}$, there exists a unique $\omega^{*}\in H^{\infty}_{1}({\mathbb{D}})$ such that

[TABLE]

Conversely, for any $\omega^{*}\in H^{\infty}_{1}({\mathbb{D}})$, if $\omega$ is given by $(\ref{eq:SChur_representation})$, then $\omega$ satisfies $\omega(z)=c_{0}+\cdots+c_{n}z^{n}+\cdots$ in ${\mathbb{D}}$. In particular, the function $\omega_{\gamma,\varepsilon}$ defined by (2.4) can be written as

[TABLE]

and $P\circ\omega_{\gamma,\varepsilon}\in{\mathcal{F}}_{\Omega}(c)$.

We next give some important properties of the Schur polynomials.

From (3.3) and (3.4) it easily follows that $\widetilde{A}_{k}(z)$ is a monic polynomial of degree $k$. Also

[TABLE]

for $k=0,1,\ldots,n$ and the degrees of $A_{k}(z),B_{k}(z)$ and $\widetilde{B}_{k}(z)$ are at most $k$.

Lemma 3.6**.**

For $k=0,\ldots,n$,

[TABLE]

Proof.

When $k=0$, (3.9) directly follows form (3.3). Assume that (3.9) holds for $k\geq 0$, then by (3.4) we have

[TABLE]

Similarly $\widetilde{B}_{k+1}(z)=z^{k+1}\overline{A_{k+1}(1/\overline{z})}$. ∎

Lemma 3.7**.**

For $k=0,\ldots,n$,

[TABLE]

Proof.

We use induction on $k$. When $k=0$, (3.10) follows directly from (3.3). Assume (3.10) holds for $k\geq 0$, then by (3.4) we have

[TABLE]

∎

Lemma 3.8**.**

For $k=0,\ldots,n$, the inequality

[TABLE]

holds for $z\in\overline{\mathbb{D}}$.

Proof.

When $k=0$, it follows from (3.3) that $|B_{0}(z)|^{2}-|A_{0}(z)|^{2}=1-|\gamma_{0}|^{2}$. Assume (3.11) holds for $k\geq 0$, then for $|z|\leq 1$, and (3.4) we have

[TABLE]

∎

Lemma 3.9**.**

For $k=0,\ldots,n$, the inequality

[TABLE]

holds for $z\in\overline{\mathbb{D}}$.

Proof.

By Lemma 3.8, the function $B_{k}(z)$ has no zeros on $\overline{\mathbb{D}}$. Hence $\widetilde{B}_{k}(z)/B_{k}(z)$ is analytic on $\overline{\mathbb{D}}$. For $|z|=1$, using Lemmas 3.6 and 3.8 we have

[TABLE]

Thus we have $|\widetilde{B}_{k}(z)/B_{k}(z)|<1$ on $\partial{\mathbb{D}}$, and hence by the maximum modulus principle for analytic functions, $|\widetilde{B}_{k}(z)/B_{k}(z)|<1$ holds on $\overline{\mathbb{D}}$. ∎

4. Proof of the Main Theorem

First we show that $V_{\Omega}^{j}(z_{0},c)$ is a compact and convex subset of ${\mathbb{C}}$.

Proposition 4.1**.**

For $c=(c_{0},\ldots,c_{n})\in\text{\rm Int}\,{\mathcal{C}}(n)$, the class ${\mathcal{F}}_{\Omega}(c)$ is a compact and convex subset of ${\mathcal{A}}({\mathbb{D}})$.

Proof.

For $g\in{\mathcal{F}}_{\Omega}(c)$, by Schwarz’s lemma we have

[TABLE]

This implies $P^{-1}(g(z))\in\overline{\Delta}(c_{0},r)=\{w\in{\mathbb{C}}:|w-c_{0}|/|1-\overline{c_{0}}w|\leq r\}(\subset{\mathbb{D}})$ for $|z|\leq r<1$. Thus $g(z)\in P(\overline{\Delta}(c_{0},r))$ for any $g\in{\mathcal{F}}_{\Omega}(c)$, and $|z|\leq r<1$. Therefore ${\mathcal{F}}_{\Omega}(c)$ is locally uniformly bounded, and hence by Montel’s theorem forms a normal family.

We next show that ${\mathcal{F}}_{\Omega}(c)$ is closed.

Let $g_{k}\in{\mathcal{F}}_{\Omega}(c)$, $k\in{\mathbb{N}}$ and $g_{0}\in{\mathcal{A}}({\mathbb{D}})$ such that $g_{k}\rightarrow g_{0}$ locally uniformly in ${\mathbb{D}}$ as $k\rightarrow\infty$. Then

[TABLE]

Thus $(P^{-1}\circ g_{0})(z)=c_{0}+c_{1}z+\cdots+c_{n}z^{n}+\cdots$ for ${z\in\mathbb{D}}$. Similarly, it follows that $g_{0}({\mathbb{D}})\subset\overline{\Omega}$. Now suppose that $g_{0}({\mathbb{D}})\backslash\Omega\not=\emptyset$. Then there exists $z^{*}\in{\mathbb{D}}$ such that $w^{*}=g_{0}(z^{*})\in\partial\Omega$. Since $g_{0}(0)=P(c_{0})\not=w^{*}=g_{0}(z^{*})$, $g_{0}$ is a non-constant analytic function and hence is an open map. Thus $g_{0}({\mathbb{D}})$ is a neighborhood of $w^{*}\in\partial\Omega$. Since there exists a support line of the convex set $\Omega$ through $w^{*}$, the neighborhood $g_{0}({\mathbb{D}})$ of $w^{*}$ contains an exterior point of $\Omega$. This contradicts $g_{0}({\mathbb{D}})\subset\overline{\Omega}$. Hence $g_{0}({\mathbb{D}})\subset\Omega$ and $g_{0}\in{\mathcal{F}}_{\Omega}(c)$, and so ${\mathcal{F}}_{\Omega}(c)$ is closed in ${\mathcal{A}}({\mathbb{D}})$. Since ${\mathcal{F}}_{\Omega}(c)$ is normal and closed in the metric space ${\mathcal{A}}({\mathbb{D}})$, it is therefore a compact subset of ${\mathcal{A}}({\mathbb{D}})$.

Next suppose $g_{0}$, $g_{1}\in{\mathcal{F}}_{\Omega}(c)$. Then $P^{-1}(g_{0}(z))-P^{-1}(g_{1}(z))=\mathcal{O}(z^{n+1})$ as $z\rightarrow 0$. Hence by Lemma 3.1, $g_{0}(z)-g_{1}(z)=\mathcal{O}(z^{n+1})$ as $z\rightarrow 0$. Thus for any $t\in[0,1]$, $g_{t}(z)=(1-t)g_{0}(z)+tg_{1}(z)$ satisfies $g_{t}(z)-g_{0}(z)=\mathcal{O}(z^{n+1})$ as $z\rightarrow 0$. Hence again using Lemma 3.1, it follows that $P^{-1}(g_{t}(z))-P^{-1}(g_{0}(z))=\mathcal{O}(z^{n+1})$, as $z\rightarrow 0$. This shows that $P^{-1}(g_{t}(z))=c_{0}+\cdots+c_{n}z^{n}+\cdots$ for ${z\in\mathbb{D}}$. Since $\Omega$ is convex, $g_{t}(z)=(1-t)g_{0}(z)+tg_{1}(z)\in\Omega$ for all $z\in{\mathbb{D}}$. Thus $g_{t}({\mathbb{D}})\subset\Omega$, and hence $g_{t}\in{\mathcal{F}}_{\Omega}(c)$. Therefore ${\mathcal{F}}_{\Omega}(c)$ is convex. ∎

Corollary 4.1**.**

For $c=(c_{0},\ldots,c_{n})\in\text{\rm Int}\,{\mathcal{C}}(n)$, the set $V_{\Omega}^{j}(z_{0},c)$ is a compact and convex subset of ${\mathbb{C}}$.

Proof.

Since the functional ${\mathcal{A}}({\mathbb{D}})\ni g\mapsto\int_{0}^{z_{0}}\zeta^{j}(g(\zeta)-g(0))\,d\zeta$ is linear and continuous on ${\mathcal{A}}({\mathbb{D}})$, and $V_{\Omega}^{j}(z_{0},c)$ is the image of the compact and convex subset ${\mathcal{F}}_{\Omega}(c)$ of ${\mathcal{A}}({\mathbb{D}})$ under this functional, $V_{\Omega}^{j}(z_{0},c)$ is also a compact and convex subset of ${\mathbb{C}}$. ∎

Proposition 4.2**.**

Let $c=(c_{0},\ldots,c_{n})\in{\rm Int}\,{\mathcal{C}}(n)$ and $z_{0}\in{\mathbb{D}}\backslash\{0\}$. Then $Q_{\gamma,j}(z_{0},0)=\int_{0}^{z_{0}}\zeta^{j}\{P(\omega_{\gamma,0}(\zeta)-P(c_{0})\}\,d\zeta$ is an interior point of the set $V_{\Omega}^{j}(z_{0},c)$, where $\gamma=(\gamma_{0},\ldots,\gamma_{n})$ is the Schur parameter of $c$.

Proof.

For fixed $z_{0}\in{\mathbb{D}}$, note that $Q_{\gamma,j}(z_{0},\varepsilon)$ defined by (2.5) is an analytic function of $\varepsilon\in\overline{\mathbb{D}}$. Since a non-constant analytic function is an open map, in order to prove the proposition it suffices to show that $Q_{\gamma,j}(z_{0},\varepsilon)$ is a non-constant function of $\varepsilon$.

Let

[TABLE]

Then the problem reduces to showing that $\psi(z)\not=0$ for $z\in{\mathbb{D}}\backslash\{0\}$. By (3.7) and Lemma 3.7 we have

[TABLE]

We note that $\psi$ has a Taylor series representation of the form $\psi(z)=az^{n+j+2}+\cdots$, with $a=(n+j+2)^{-1}\prod_{k=0}^{n}(1-|\gamma_{k}|^{2})P^{\prime}(c_{0})$. The assertion will be proved provided we can show that $\mbox{\rm Re}\,(z\psi^{\prime\prime}(z)/\psi^{\prime}(z))\geq 0$, since in this case, Lemmas 3.3 and (3.8) imply that there exists a starlike univalent function $\psi_{0}\in\mathcal{S}^{*}$ satisfying $\psi(z)=a\psi_{0}(z)^{n+j+2}$, and so $\psi(z)$ has no zeros in ${\mathbb{D}}\backslash\{0\}$.

In order to show $\text{\rm Re}\,(z\psi^{\prime\prime}(z)/\psi^{\prime}(z))\geq 0$, by Lemma 3.4 we may assume without loss of generality that

[TABLE]

where $\eta_{i}\in\partial{\mathbb{D}}$, $0<\beta_{i}\leq 2$ $(i=1,2,\ldots,m)$, and $\sum_{i=1}^{m}\beta_{i}=2$. Under this assumption

[TABLE]

and so

[TABLE]

For $i=1,\ldots,m$, let $z_{i1},\ldots,z_{ip_{i}}$ $(0\leq p_{i}\leq n)$ be zeros of $B_{n}(z)-\eta_{i}\widetilde{B}_{n}(z)$. Then by Lemma 3.9, we have $|z_{i\ell}|>1$ for all $i=1,\ldots m$ and $\ell=1,\ldots,p_{i}$. Since $B_{n}(0)=1$ and $\widetilde{B}_{n}(0)=\gamma_{0}$, it follows that $B_{n}(z)-\eta_{i}\widetilde{B}_{n}(z)=(1-\eta_{i}\gamma_{0})\prod_{\ell=1}^{p_{i}}(1-z/z_{i\ell})$. Thus using the identity $w/(1-w)=2^{-1}\{(1+w)/(1-w))-1\}$ we have

[TABLE]

∎

Proposition 4.3**.**

Let $c=(c_{0},\ldots,c_{n})\in{\rm Int}\,{\mathcal{C}}(n)$ and $z_{0}\in{\mathbb{D}}\backslash\{0\}$. Then

[TABLE]

holds for all $\varepsilon\in\partial{\mathbb{D}}$, where $\gamma$ is the Schur parameter of $c$. Furthermore,

[TABLE]

for some $g\in{\mathcal{F}}_{\Omega}(c)$, and $\varepsilon\in\partial{\mathbb{D}}$ if, and only if, $g(z)\equiv P(\omega_{\gamma,\varepsilon}(z))$.

Proof.

Let $g\in{\mathcal{F}}_{\Omega}(c)$ and $\omega=P^{-1}\circ g$. Then by Lemma 3.5, there exists $\omega^{*}\in H_{1}^{\infty}({\mathbb{D}})$ such that

[TABLE]

where $A_{n}(z)$, $B_{n}(z)$, $\widetilde{A}_{n}(z)$ and $\widetilde{B}_{n}(z)$ are the Schur polynomials associated with $\gamma$. Since $\omega^{*}\in H_{1}^{\infty}({\mathbb{D}})$, we have

[TABLE]

It follows from Lemmas 3.7 and 3.8 that the inequality (4.1) is equivalent to

[TABLE]

where

[TABLE]

Thus $(P^{-1}\circ g)(z)=\omega(z)\in\overline{\mathbb{D}}(\rho(z),r(z))$, and so $g(z)\in P(\overline{\mathbb{D}}(\rho(z),r(z)))$ for any $g\in{\mathcal{F}}_{\Omega}(c)$. Since a convex univalent function maps any closed subdisk of ${\mathbb{D}}$ onto a convex closed Jordan domain with an analytic convex boundary curve, for any $z\in{\mathbb{D}}\backslash\{0\}$ and $\theta\in{\mathbb{R}}$, $g(z)$ belongs to the left half plane of the tangential line at $P(\rho(z)+r(z)e^{i\theta})$ with the tangential vector $ir(z)e^{i\theta}P^{\prime}(\rho(z)+r(z)e^{i\theta})$ (see Figure 1). Thus

[TABLE]

Since the tangential line intersects the boundary curve only at $P(\rho(z)+r(z)e^{i\theta})$, equality in (4.2) holds if, and only if, $g(z)=P(\rho(z)+r(z)e^{i\theta})$.

If $g(z)=(P\circ\omega_{\gamma,\varepsilon})(z)$ with $|\varepsilon|=1$, Lemma 3.5 shows that

[TABLE]

Hence by Lemma 3.7

[TABLE]

In particular, $|\omega_{\gamma,\varepsilon}(z)-\rho(z)|=r(z)$. Thus for any $z\in{\mathbb{D}}\backslash\{0\}$, there exists $\theta\in{\mathbb{R}}$ such that $\omega_{\gamma,\varepsilon}(z)=\rho(z)+r(z)e^{i\theta}$. Substituting this into (4.2) we have

[TABLE]

with equality if, and only if, $g(z)=P(\omega_{\gamma,\varepsilon}(z))$. Rewriting (4.4) and using (4.3) we obtain

[TABLE]

for $z\in{\mathbb{D}}\backslash\{0\}$, and so by Lemma 3.8 we have

[TABLE]

for $z\in{\mathbb{D}}\backslash\{0\}$, where

[TABLE]

We now claim that

[TABLE]

Assuming the inequality (4.6) for the moment, we complete the proof.

Notice that $h$ has a Taylor series representation of the form $h(z)=az^{n+j+2}+\cdots$, with $a=(n+j+2)^{-1}P^{\prime}(c_{0})$. By Lemma 3.3, there exists $h_{0}\in\mathcal{S}^{*}$ such that $h(z)=ah_{0}(z)^{n+j+2}$. Since $h_{0}$ is starlike, for any $z_{0}\in\mathbb{D}\backslash\{0\}$, the line segment joining [math] and $h_{0}(z_{0})$ lies entirely in $h_{0}(\mathbb{D})$. Define a curve $\Gamma:z=z(t)$, $0\leq t\leq 1$ joining [math] to $z_{0}$ by $z(t)=h_{0}^{-1}(t^{1/(n+j+2)}h_{0}(z_{0}))$. Then

[TABLE]

Thus $h^{\prime}(z(t))z^{\prime}(t)\equiv h(z_{0})$ for $0<t\leq 1$. This, and (4.5), shows that

[TABLE]

This implies that for any $g\in{\mathcal{F}}_{\Omega}(c)$, the value of the integral $\int_{0}^{z_{0}}z^{j}\{g(z)-g(0)\}\,dz$ belongs to a closed half plane, i.e.,

[TABLE]

where $w_{0}=Q_{\gamma,j}(z_{0},\varepsilon)$ and $\alpha=\varepsilon h(z_{0})$. Thus

[TABLE]

Since $P\circ\omega_{\gamma,\varepsilon}\in{\mathcal{F}}_{\Omega}(c)$, it follows that

[TABLE]

and so from (4.8) and (4.9) we obtain $Q_{\gamma,j}(z_{0},\varepsilon)=w_{0}\in\partial V_{\Omega}^{j}(z_{0},c)$.

We next deal with uniqueness. Suppose that

[TABLE]

holds for some $g\in{\mathcal{F}}_{\Omega}(c)$ and $\varepsilon\in\partial{\mathbb{D}}$. Then from (4.7) it follows that

[TABLE]

holds on the curve $\Gamma$, and so

[TABLE]

on $\Gamma$. By the equality condition of (4.4), we have $g(z)=P(\omega_{\gamma,\varepsilon}(z))$ on $\Gamma$, and from the identity theorem for analytic functions it follows that $g=P\circ\omega_{\gamma,\varepsilon}$.

We finally prove (4.6).

Since the harmonic function ${\rm Re}(zh^{\prime\prime}(z)/h^{\prime}(z))+1$ assumes the value $n+j+2\geq 1>0$ at the origin, by using the minimum principle it suffices to show $\text{\rm Re}(zh^{\prime\prime}(z)/h^{\prime}(z))+1\geq 0$ in ${\mathbb{D}}$.

As in the proof of Proposition 4.2, we may assume without loss of generality that

[TABLE]

where $\eta_{i}\in\partial{\mathbb{D}}$, $0<\beta_{i}\leq 2$ $(i=1,2,\ldots,m)$, and $\sum_{i=1}^{m}\beta_{i}=2$. Thus from Lemma 3.5 we have

[TABLE]

and hence a simple computation gives

[TABLE]

We note that for each $|\varepsilon|=1$, the function $\omega_{\gamma,\varepsilon}(z)=\{\widetilde{B}_{n}(z)+\varepsilon z\widetilde{A}_{n}(z)\}/\{B_{n}(z)+\varepsilon zA_{n}(z)\}$ is a finite Blaschke product. Thus for each $i$, the polynomials $q_{i}(z):=B_{n}(z)+\varepsilon zA_{n}(z)-\eta_{i}\{\widetilde{B}_{n}(z)+\varepsilon z\widetilde{A}_{n}(z)\}$ have no zeros in ${\mathbb{D}}$, and are of degree $n+1$ at most. Since $B_{n}(0)=1$ and $\widetilde{B}_{n}(0)=\gamma_{0}$, the polynomials $q_{i}(z)$ can be expressed as

[TABLE]

where $z_{i1},\ldots,z_{ip_{i}}\in{\mathbb{C}}\backslash{\mathbb{D}}$ are the zeros of the polynomial $q_{i}(z)$, and $0\leq p_{i}\leq n+1$. Therefore using the identity $w/(1-w)=2^{-1}\{(1+w)/(1-w)-1\}$ we have

[TABLE]

∎

Now we are in a position to prove our main result.

Proof of Theorem 2.2.

Let $c\in\mbox{\rm Int}\,{\mathcal{C}}(n)$, $z_{0}\in{\mathbb{D}}\backslash\{0\}$ and $\gamma=(\gamma_{0},\ldots,\gamma_{n})$ be the Schur parameter of $c$. Corollary 4.1 and Proposition 4.2, shows that the set $V_{\Omega}^{j}(z_{0},c)$ is a compact and convex subset of ${\mathbb{C}}$, and $Q_{\gamma,j}(z_{0},0)$ is an interior point of $V_{\Omega}^{j}(z_{0},c)$. From these properties it is not difficult to see that $\partial V_{\Omega}^{j}(z_{0},c)$ is a Jordan curve, and $V_{\Omega}^{j}(z_{0},c)$ is a union of $\partial V_{\Omega}^{j}(z_{0},c)$ and its inside domain, i.e., $V_{\Omega}^{j}(z_{0},c)$ is a closed Jordan domain.

Proposition 4.3 shows that the map $\partial{\mathbb{D}}\ni\varepsilon\mapsto Q_{\gamma,j}(z_{0},\varepsilon)\in\partial V_{\Omega}^{j}(z_{0},c)$ is a closed curve. Furthermore, it is a simple curve. Indeed, if $Q_{\gamma,j}(z_{0},\varepsilon_{1})=Q_{\gamma,j}(z_{0},\varepsilon_{2})$ for some $\varepsilon_{1},\varepsilon_{2}\in\partial{\mathbb{D}}$, then by the uniqueness part of Proposition 4.3 we have $P(\omega_{\gamma,\varepsilon_{1}}(z))\equiv P(\omega_{\gamma,\varepsilon_{2}}(z))$, and so $\omega_{\gamma,\varepsilon_{1}}(z)\equiv\omega_{\gamma,\varepsilon_{2}}(z)$. Using the representation (3.7) for $\omega_{\gamma,\varepsilon}$, we have

[TABLE]

which gives

[TABLE]

Consequently, by Lemma 3.7 we conclude that $\varepsilon_{1}=\varepsilon_{2}$.

Since a simple closed curve cannot contain any simple closed curve other than itself, the map $\partial{\mathbb{D}}\ni\varepsilon\mapsto Q_{\gamma,j}(z_{0},\varepsilon)\in\partial V_{\Omega}^{j}(z_{0},c)$ is surjective, and a parametrization of the boundary curve $\partial V_{\Omega}^{j}(z_{0},c)$. It therefore follows from Darboux’s theorem (see [9, Lemma 1.1]) that for fixed $z_{0}\in{\mathbb{D}}\backslash\{0\}$, $Q_{\gamma,j}(z_{0},\varepsilon)$ is a convex univalent analytic function of $\varepsilon\in\overline{\mathbb{D}}$, and $V_{\Omega}^{j}(z_{0},c)=\{Q_{\gamma,j}(z_{0},\varepsilon):\varepsilon\in\overline{\mathbb{D}}\}$. This completes the proof. ∎