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\begin{document}
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\Year{2008}
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\keywords{omitted area problem, logarithmic capacity, univalent function, symmetrization, local variation}
\subjclass[msc2000]{30C70}
\title[Iceberg-type Problems]{ Iceberg-type Problems: Estimating Hidden Parts of a Continuum from the Visible Parts} %
%% Please do not enter footnotes or \inst{}-notes into the optional
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\author[R.W. Barnard]{Roger W. Barnard
\footnote{E-mail: \scriptsize{\textsf{roger.w.barnard@ttu.edu}},
Phone: \,806\,742\,2566,
Fax: \,806\,742\,1112}}
%% Information for the second author
\author[K. Pearce]{Kent Pearce
\footnote{Corresponding author:
E-mail: \scriptsize{\textsf{kent.pearce@ttu.edu}},
Phone: \,806\,742\,2566,
Fax: \,806\,742\,1112}}
%%
%% Information for the third author
\author[A.Y. Solynin]{Alexander Yu. Solynin
\footnote{E-mail: \scriptsize{\textsf{alex.solynin@ttu.edu}},
Phone: \,806\,742\,2566,
Fax: \,806\,742\,1112}
\footnote{Partially supported by NSF grant DMS-0525339}}
\address{Department of Mathematics and Statistics, Texas Tech
University, Box 41042, Lubbock, TX 79409}
\begin{abstract}{We consider the complex plane $\mathbb{C}$ as a space
filled by two different media, separated by the real axis
$\mathbb{R}$. We define $\mathbb{H}_+=\{z: \,\Im \,z>0\}$ to be the upper
half-plane. For a planar body $E$ in $\mathbb{C}$, we discuss a problem of
estimating characteristics of the ``invisible'' part,
$E_-=E\setminus \mathbb{H}_+$, from characteristics of the whole
body $E$ and its ``visible'' part, $E_+=E\cap \mathbb{H}_+$. In this
paper, we find the maximal draft of $E$ as a function of the
logarithmic capacity of $E$ and the area of $E_+$.
%AMS No. 30
}
\end{abstract}
\maketitle
%----- Section 1: Introduction --------------------------------------
\section{Introduction} \label{Introduction}
We will discuss problems (called {\it iceberg-type problems} below) of estimating
characteristics of the ``invisible" part of a compact set $E$ in the complex plane
$\mathbb{C}$ from some known characteristics of the whole set and its ``visible" part.
We emphasis from the beginning that the problems we study in this paper are not directly
related to (real) physical icebergs. The problem name reflects the fact that the
object under consideration consists of two parts, hidden and visible, and the question
is to recover some of the properties of the hidden part from the visible part. During
the ages a titanic work has been done to solve this problem in its
everyday physical setting.
In this paper, we study iceberg-type problems in two-dimensional
space, which will be identified as the complex plane $\mathbb{C}$.
Accordingly, $\overline{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$,
$\mathbb{H}_+=\{z:\,\Im\,z> 0\}$, and $\mathbb{H}_-=\{z:\,\Im\,z<0\}$
will denote the extended complex plane, the upper half-plane, and
the lower half-plane, respectively. The real axis $\mathbb{R}$
will play the role of the surface of interface between
$\mathbb{H}_+$ and $\mathbb{H}_-$.
%\smallskip
For any given compact set $E$ in $\mathbb{C}$, we define $E_+=E\cap
\mathbb{H}_+$ and $E_-=E\cap \overline{\mathbb{H}}_-$. The sets $E_+$ and $E_-$ denote the
visible and hidden parts of $E$, respectively.
%\smallskip
An accumulative characteristic of any body $E$ surrounded by media
is its potential or capacity. In our two-dimensional setting, the
\textit{logarithmic capacity} will be chosen as
the primary characteristic of $E$. We remind the reader that
the logarithmic capacity, $\CAP E$, of a compact set $E$ is defined
by %
$$ %
-\log \CAP E =\lim_{z\to \infty}(g(z)-\log|z|), %
$$ %
where $g(z)$ denotes Green's function of the unbounded component
$D(E)$ of $\overline{\mathbb{C}}\setminus E$ having singularity at
$z=\infty$. Let ${\mathcal{F}}$ be the collection of all continua
($=$ connected compact
sets) $E$ in $\mathbb{C}$ such that %
$$ %\be \label{1.1} %
\CAP(E)=1. %
$$ %\ee %
\smallskip
For the measured characteristic of a visible part $E_+$ we will choose
the mass of $E_+$, which, assuming homogeneity of $E$, is
proportional to the area of $E_+$. For $E$ in ${\mathcal{F}}$, the
well known estimates of the
logarithmic capacity show that %
$$ %\be \label{1.2} %
0\le \Area(E_+)\le \Area(E)\le \pi (\CAP (E))^2=\pi. %
$$ %\ee %
Characteristics of the hidden part $E_-$ which one may want to
control and which are of a particular importance, include: the
draft of the iceberg $H(E)$, the width of the invisible part of
the iceberg $w(E)$, and the safe distance from the iceberg $d(E)$.
Figure~\ref{Figure_Iceberg} illustrates these characteristics
while the precise definitions are as follows:
\be \label{equation draft} %
H(E)=\max \,(-\Im (z)), %
\ee %
where the maximum is taken over all $z$ in $E$, %
\be \label{equation width} %
w(E)=\max \,(\Re (z_2-z_1)), %
\ee %
where the maximum is taken
over all $z_1,z_2$ in $E_-$, and %
\be \label{equation safe} %
d(E)=\max \,(\Re (z_2))-\sup\,(\Re(z_1)), %
\ee %
where the maximum is taken over all $z_2$ in $E_-$ and the supremum
is taken over all $z_1$ in $E_+$.
\smallskip
\begin{figure}
$$\includegraphics[scale=.3,angle=-90]{iceberg-boat} $$
\caption{Two-dimensional iceberg.}
\label{Figure_Iceberg}
\end{figure}
\smallskip
Then, the extremal problem for each of the functionals
(\ref{equation draft}), (\ref{equation width}), and (\ref{equation safe})
is to find its maximal value over the class ${\mathcal{F}}$
and describe all possible extremal continua. We define %
\be \label{equation maximals} %
{\textbf{(a)}} \ \ \ H({\mathcal{F}})=\max H(E) \quad \quad
{\textbf{(b)}} \ \ \ w({\mathcal{F}})=\max w(E)
\quad \quad {\textbf{(c)}} \ \ \ d({\mathcal{F}})=\max d(E), %
\ee %
where in each case the maximum is taken over all sets $E$ in
${\mathcal{F}}$.
Our main goal in this paper is to give a complete solution to
problem (\ref{equation maximals})\textbf{(a)}. Problems
(\ref{equation maximals})\textbf{(b)} and
(\ref{equation maximals})\textbf{(c)} along
with some other questions will be discussed in the last section.
\medskip
As is well known, problems on the logarithmic capacity of continua
can be reformulated as problems about functions in the class $\Sigma'$ of univalent functions %
\be \label{equation Sigma} %
f(z)=z^{-1}+a_0+a_1z+\cdots, %
\ee %
which are analytic in the unit disk $\mathbb{D}$, except for a
simple pole at $z=0$. For $f$ in $\Sigma'$, define
$E_f=\overline{\mathbb{C}}\setminus f(\mathbb{D})$ and define
$\Sigma'_0=\{f\in \Sigma':\,0\in E_f\}$.
\smallskip
We will solve problem (\ref{equation maximals})\textbf{(a)} by
solving its reformulated dual problem for the class $\Sigma'_0$.
There is a technical advantage in shifting to the dual problem in that
the analytical and constructional difficulties which surround the dual
problem are more tractable than those in the original setting. The
precise formulation of the dual of the
maximal draft problem (\ref{equation maximals})\textbf{(a)} is the
following problem on the maximal omitted area for the class
$\Sigma'_0$.
For any given real $h$ such that $0h\}), %
\ee %
where the maximum is taken over all $f$ in $\Sigma'_0$, and find all
functions $f$ in $\Sigma'_0$ extremal for (\ref{equation dual}).
Thus, the question is, for any given $h$, such that $0h\}$.
We note here that our parameter $h$, which is equal to
the horizontal distance from $w=0$ to the half-plane
$\mathbb{H}_h$, gives as well the value of the maximal draft of
icebergs with visible area $A=A(h)$. In addition, in Corollary~\ref{Corollary
1} (below) we show that the extremal configuration for problem
(\ref{equation maximals})\textbf{(a)} coincides with the extremal
configuration for problem (\ref{equation dual}) up to rotation and translation.
For convenience we define $A_f(h) = \Area (E_f\cap \mathbb{H}_h)$.
The maximal omitted area problem (\ref{equation dual}) is solved by the
following theorem. %
\bt \label{Theorem 1} %
Let $h$ satisfy $00\}, %
\ee %
maps the semidisk \, ${\mathbb{D}}_+$ conformally onto the unit
disk
$\mathbb D$ and %
\be \label{equation Thm1-F} %
F(s) = -2\beta r(1-r^2)\int_0^s \frac{t(t^2+1)\sqrt{1+\tau^2
t^2}}{(t^2-r^2)^2\sqrt{t^2+\tau^2}}\,dt %
\ee %
with the principal branches of the radicals and
with $\tau$ and $\beta$ defined by (\ref{equation Thm1-C}) and (\ref{equation Thm1-D}).
\et %
\smallskip
Theorem~\ref{Theorem 1} shows that the maximal omitted area $A(h)$
is given by the explicit expression in the right-hand side of
(\ref{equation Thm1-A}) with $r$, $\tau$, and $\beta$ defined by
(\ref{equation Thm1-B}), (\ref{equation Thm1-C}), and (\ref{equation Thm1-D}),
respectively. Its graph shown in Figure~\ref{Figure_Extremal_Area}
suggests, and we will prove this in Lemma~\ref{Lemma 2.1} in
Section~\ref{Section 2}, that $A(h)$ strictly decreases from $\pi$
to $0$ as $h$ runs from $0$ to $4$. Therefore, the inverse,
$h=\Psi(A)$, of the function $A(h)$ is well defined on $0\le A\le
\pi$.
\bc \label{Corollary 1} %
If $E\in {\mathcal{F}}$ has the visible area $A$, $00$;
\textbf{(e)} $|f'(z)|\to \beta$ as
$z\to e^{i\theta_1}$ such that $z\in \overline{\mathbb{D}}$. %
\el %
\textit{Proof.} %
\textbf{(a)} First we prove that $f'$ is bounded near $l_{fr}$. If
not then there is $e^{i\theta_0}$ in $l_{fr}$ and a sequence $z_k\to
e^{i\theta_0}$ such that $z_k\in \mathbb{D}$ for all $k$ in
$\mathbb{N}$ and $f'(z_k)\to \infty$.
Let $\varphi_k$ denote the conformal mapping from $\mathbb{D}$
onto the domain $\mathbb{D}\setminus
\overline{\mathbb{D}}_{\varepsilon_k}(z_k)$ with
$\varepsilon_k=1-|z_k|$ normalized by $\varphi_k(0)=0$,
$\varphi'_k(0)>0$ and define $f_k=\beta_k f \circ \varphi_k$ with
$\beta_k=1-\pi^2\varepsilon^2_k/6$. One can easily verify (see,
for example, Lemma~3.1 in \cite{BS}) that $f_k\in \Sigma'_0$.
Since $00$ independent of $f$. Since $f'(z_k)\to
\infty$ as $k\to \infty$, (\ref{equation AreaA}) contradicts the
extremality of $f$. Therefore, $f'$ is bounded near $l_{fr}$.
\smallskip
\textbf{(b)} First we show that $|f'(z)|$ is constant a.e.\ on
$l_{fr}$. Since $L_{fr}$ is Jordan locally rectifiable, it follows
that the
non-zero finite limit %
\be \label{equation Lem2-b-1} %
f'(\zeta)=\lim_{z\to\zeta,z\in \overline{\mathbb{D}}}
\frac{f(z)-f(\zeta)}{z-\zeta}\not= 0,\infty %
\ee %
exists a.e. on $l_{fr}$; see \cite[Theorem 6.8, Exercise
6.4.5]{P}. Assume that %
\be \label{equation Lem2-b-2}%
0<\beta_1=|f'(e^{i\nu_1})|<|f'(e^{i\nu_2})|=\beta_2<\infty %
\ee %
for $e^{i\nu_1}, e^{i\nu_2}\in l_{fr}$. Note that (\ref{equation
Lem2-b-1}) and (\ref{equation Lem2-b-2}), combined with the fact
that $f'$ is bounded near $l_{fr}$, allow us to apply the
two-point variational formulas, see \cite[Lemma~10]{BS} or
\cite[Lemma~5]{BPS1}. Namely, for fixed positive $k_1$, $k_2$ such
that $ 0k_2\beta_2^{-1}$ and
fixed $\varphi>0$ small enough, we consider the two-point
variation $\tilde D$ of $D=f(\mathbb{D})$ centered at
$w_1=f(e^{i\nu_1})$ and $w_2=f(e^{i\nu_2})$ with inclinations
$\varphi$ and radii $\e_1=k_1 \e$, $\e_2=k_2 \e$, respectively;
see \cite[Section~2]{BPS1}. Computing the change in the area by
\cite[formula (2.11)]{BPS1}, we find %
\be \label{equation Lem2-b-3}%
{\rm{Area}}\,(\mathbb{C}\setminus\tilde D)-
{\rm{Area}}\,E_f=\frac{2\pi\varphi-\sin 2\pi\varphi}{2\sin^2
\pi\varphi}\e^2(k_1^2-k_2^2)+o(\e^2)>0 %
\ee %
for all $\e>0$ small enough. Similarly, applying \cite[formula
(2.10)]{BPS1}, we get %
\be \label{equation Lem2-b-4} %
\frac{\CAP(\mathbb{C}\setminus \tilde
D)}{\CAP(E_f)}=\left[\frac{\varphi(2+\varphi)}{6(1+
\varphi)^2}\frac{k_1^2}{\beta_1^2}-
\frac{\varphi(2-\varphi)}{6(1-\varphi)^2}\frac{k_2^2}{\beta_2^2}\right]
\e^2+o(\e^2)<0 %
\ee %
for all $\e>0$ small enough and $\varphi$ chosen such that the
expression in the brackets is positive.
Inequalities (\ref{equation Lem2-b-3}) and (\ref{equation Lem2-b-4}) lead to
a contradiction to the extremality of $f$ for $A(h)$, via a
standard subordination argument.
Thus $|f'(e^{i\theta})|=\beta$ a.e.\ on $l_{fr}$ with some $\beta>0$.
To prove that $|f'(e^{i\theta})|=\beta$ everywhere on $l_{fr}$, we consider the
auxiliary conformal mapping %
\be \label{equation 2.7} %
g=\varphi\circ f\circ k_\tau \quad \quad {\mbox{ with
\ \ $\varphi(w)=1/(w-p_h)$,}} %
\ee %
where $p_h$ is defined in the proof of
Lemma~\ref{Lemma 2.1}, and with %
$$ %\be \label{equation 2.4} %
k_\tau(\zeta)=k^{-1}(\tau k(\zeta)), \quad {\mbox{ where
$k(\zeta)=\zeta/(1-\zeta)^2$ and $\tau=1/\sin^{2}(\theta_2/2)$.}} %
$$ %
We note that $k_\tau$ maps the slit disk
$\mathbb{D}'=\mathbb{D}\setminus[-1,-r_0]$, where
$r_0=(\sqrt{\tau}-\sqrt{\tau-1})^2$, conformally and one-to-one
onto $\mathbb{D}$ in such a way that the radial slit is mapped
onto the arc $l_h=\{e^{i\theta}:\,|\theta-\pi|\le \pi-\theta_2\}$.
Let $D_g'=g(\mathbb{D}')$ and let $D_g=D'_g\cup
((p_h-h)^{-1},-p_h^{-1}]$. By the Schwarz reflection principle,
the function $g$ can be continued to a function, still denoted by
$g$, which maps the whole disk $\mathbb{D}$ conformally and
one-to-one onto $D_g$. It follows from Lemma~\ref{Lemma
2.1}\textbf{(c)} that $D_g$ is a bounded Jordan domain, whose
boundary satisfies the Lavrent'ev condition (\ref{equation Lem1})
for some $C>0$. Therefore, $D_g$ is a Smirnov domain; see
\cite[Sections~7.3, 7.4]{P}.
Thus, $\log |g'|$ can be represented by the Poisson integral %
\be \label{equation Poisson}%
\log |\varphi'(w)f'(z)k'_\tau(\zeta)|=\log|g'(\zeta)|=\frac{1}{2\pi}\int_0^{2\pi}P(r,\psi-t)%
\log|g'(e^{it})|\, dt %
\ee %
with boundary values defined a.e. on $\T$; see \cite[p. 155]{P}.
Equation (\ref{equation Poisson}) easily implies that
$$|g'(e^{i\psi})|=\beta |\varphi'(f(k_\tau(e^{i\psi}))||k'_\tau(e^{i\psi})|$$
for all $e^{i\psi}$ such that $k_\tau(e^{i\psi})\in l_{fr}$ and therefore
$|f'(e^{i\theta})|=\beta$ for all $e^{i\theta}\in l_{fr}$. In
addition, (\ref{equation Poisson}) implies that $\log f'$ is
bounded on $\overline{\mathbb{D}}$ outside any neighborhoods of
the points $z=0$, $z=-1$, and $z=e^{\pm i\theta_2}$.
\smallskip
\textbf{(c)} Since $E_f$ is Steiner symmetric w.r.t.\ $\R$, the
strict monotonicity of $|f'|$ along $l_v^+$ follows from
\cite[Lemma~4]{BS}. To prove that $|f'(e^{i\theta})|>\beta$ for
all $e^{i\theta}\in l_v\setminus\{0\}$, we assume that
$\beta=|f'(e^{i\nu_1})|>|f'(e^{i\nu_2})|=\beta_2$ with
$e^{i\nu_1}\in l_{fr}$ and some $e^{i\nu_2}\in l_v^+$. Then,
applying the two-point variation as above, we get inequalities
(\ref{equation Lem2-b-3}) and (\ref{equation Lem2-b-4}), contradicting the
extremality of $f$ for $A(h)$, again via a subordination argument.
Hence, $|f'(e^{i\theta})|\ge \beta$ for all $e^{i\theta}\in l_v$
which, when combined with the strict monotonicity property of
$|f'|$, leads to the strict inequality $|f'(e^{i\theta})|>\beta$
for $e^{i\theta}\in l_v$.
\smallskip
\textbf{(d)} Assume that $a_f=0$. Then, $\theta_1=\theta_2$,
$L_{nf}=I(h)$, and $\hat L=L_{fr}\cup \{h\}$. In addition,
$|f'(e^{i\theta})|=\beta>0$ for all $e^{i\theta}\in l_{fr}$ by
part~\textbf{(b)} of this proof.
In the notation of part \textbf{(b)}, we consider the function
$g=\varphi\circ f\circ k_\tau$ defined by (\ref{equation 2.7}),
which maps $\mathbb{D}$ conformally onto the domain $D_g$. As we
have mentioned above,
$\log |g'(\zeta)|$ can be represented
by the Poisson integral (\ref{equation Poisson}).%
Since $|k'_\tau(e^{i\psi})|\to 0$ as $\psi\to \pi$, it follows
that
$|g'(e^{i\psi})|=\beta|\varphi'(k_\tau(e^{i\psi}))||k'_\tau(e^{i\psi})|\to
0$ as $\psi\to \pi$. Therefore, %
\be \label{equation 2.5} %
\log|g'(e^{i\psi})|\to -\infty \quad \quad {\mbox{as $\psi\to
\pi$.}} %
\ee %
From (\ref{equation Poisson}) and (\ref{equation 2.5}), using the
well-known properties of the radial limits of the Poisson
integral, we obtain that %
\be \label{equation 2.6} %
\log|g'(-r)|\to -\infty \quad \quad {\mbox{as $r\to 1^-$.}} %
\ee %
Now we show that $g$ has a finite non-zero angular derivative at
$\zeta=-1$. To do this, we construct two comparison functions
$f_1$ and $f_2$. Let $f_1$ map $\mathbb{D}$ conformally onto the
vertical strip $\{w:\,0<\Re\,w0$.
\textbf{(e)} To show that $|f'|$ is continuous at $e^{\pm
i\theta_1}$, we again use the function $g$ defined by
(\ref{equation 2.7}). Using Theorem~4.14 in \cite{P} with $g_1$
defined in part \textbf{(d)} of this proof as a comparison
function, we conclude that the finite angular derivative
$g'(k_\tau^{-1}(e^{i\theta_1}))$, and therefore the angular
derivative $f'(e^{i\theta_1})$, exists finitely.
By the reflection principle, $f$ can be continued analytically
across $l_v^-$. By Lemma~\ref{Lemma 2.1}, $E_f$ is Steiner
symmetric w.r.t. $\mathbb{R}$ and circularly symmetric w.r.t.
$\mathbb{R}_0$. Using these facts it is not difficult to see that
this analytic continuation, say $\tilde f$, of $f$ is univalent in
the disk
$U=\{z:\,|z-\varepsilon_0e^{i(\theta_1+\theta_2)/2)}|<\rho_0\}$
for a sufficiently small positive $\varepsilon_0$ and
$\rho_0=|e^{i\theta_1/2}-\varepsilon_0e^{i\theta_2/2}|$. By
Proposition~4.9 \cite{P}, the function $\tilde f$ has the angular
derivative ${\tilde f}'(e^{i\theta_1})$ at $z=e^{i\theta_1}$,
which of course coincides with the angular derivative
$f'(e^{i\theta_1})$.
We have $l_v^-\subset U$. Since $|f'(e^{i\theta})|$ is monotone
and greater than $\beta$ on $l_v^-$, it follows that
$\lim_{\theta\to \theta_1^+}
f'(e^{i\theta})=\beta_0e^{-i\theta_1}$ where
$0 < \beta \le \beta_0$. Therefore, %
\be \label{equation 2.8} %
f'(z)\to \beta_0e^{-i\theta_1} \quad \quad {\mbox{as \ \ $z\to
e^{i\theta_1}$}} %
\ee %
in any Stolz angle in $\mathbb{D}$ with the vertex at
$e^{i\theta_1}$. To show that $\beta_0=\beta$, we use the Poisson
integral (\ref{equation Poisson}). Let
$\psi_1=\arg(k_\tau^{-1}(e^{i\theta_1}))$. If $\beta_0\not=\beta$,
then the theorem about radial limits of the Poisson integral
implies
that %
$$ %
\lim_{r\to 1^-}\log|g'(re^{i\psi_1})|=\frac12\lim_{\varepsilon \to
0}
\log|g'(e^{i(\psi_1+\varepsilon)})g'(e^{i(\psi_1-\varepsilon)})|. %
$$ %
This implies that $|f'(k_\tau(re^{i\theta_1}))|\to
\sqrt{\beta\beta_0}$ as $r\to 1^-$, which together with
(\ref{equation 2.8}) shows that we must have $\beta_0=\beta$.
Using the Poisson integral (\ref{equation Poisson}) once more, we
conclude that $\log|g'(\zeta)|$ is continuous for $\zeta$ such
that $|\zeta|\le 1$ and $|\zeta-k_\tau^{-1}(e^{i\theta_1})|$ is
small enough. Since $g=\varphi\circ f\circ k_\tau$ and $\varphi$
and $k_\tau$ are conformal in the corresponding domains the latter
implies \textbf{(e)}.
The proof of Lemma~\ref{Lemma 2.2} is complete. \hfill $\Box$
\section{Closed form of the extremal functions and the proof of Theorem~\ref{Theorem 1}}
\label{Section 3} %
\setcounter{equation}{0} %
Lemmas \ref{Lemma 2.1} and \ref{Lemma 2.2} provide sufficient
information to find a closed form of the function $f$ extremal for
$A(h)$ when $00$ such that %
\be \label{equation Lem3} %
F'_r(s)=-\frac{2\beta
r(1-r^2)s(s^2+1)(1+\tau^2s^2)^{1/2}}{(s^2-r^2)^2(s^2+\tau^2)^{1/2}} %
\ee %
with the principal branches of the radicals. %
\el %
\textit{Proof.} Let $\theta_1$ and $\theta_2$ be the angles
defined for $f$ as in Section~\ref{Section 2}. Since $\theta(r)$
defined by (\ref{equation arcsin}) strictly increases in $0 0$. We can write %
\be \label{equation Q1} %
\frac{Q_1}{3/2-Q_1} =
\frac{(4+4r^2)P-(4+4r^4)}{(9r^2+10+7r^4)-(7+7r^2)P}. %
\ee %
It is easily seen from (\ref{equation P and Q}) that $1 < P <
\sqrt{2}$ for $0 < r < 1$. Hence, it is clear that the numerator
in (\ref{equation Q1}) is positive. On the other hand, we have %
$$ %
(9r^2+10+7r^4)^2-(7+7r^2)^2P^2 = 270r^4+82r^2+126r^6+2
> 0, %
$$
which shows that the denominator in (\ref{equation Q1}) is
positive as well. The lemma is proved. \hfill $\Box$
\medskip
All the results established so far were used to prove
Theorem~\ref{Theorem 1}. Now we prove monotonicity properties of
the functions $\tau=\tau(r)$, $\beta=\beta(r)$, and $a=a(r)$.
Although not needed for our main proof they provide some
additional information about extremal configurations. The graph of
$a(r)$ is displayed in Figure~\ref{Figure_h_and_a} and the graphs
of functions $\tau(r)$ and $\beta(r)$ are displayed in Figure~\ref{Figure_tau_and_beta}.%
\begin{figure}
$$
\includegraphics[scale=.35,angle=0]{graph-of-tau-alt}
\includegraphics[scale=.35,angle=0]{graph-of-beta-alt}
$$
\caption{Functions $\tau(r)$ and $\beta(r)$.}
\label{Figure_tau_and_beta}
\end{figure}
\medskip
\bl \label{Lemma tau} %
The function $\tau=\tau(r)$ defined by (\ref{equation Thm1-C}) strictly decreases
from $\sqrt{\sqrt 2-1}$ to $0$ as $r$ runs from $0$ to $1$. %
\el %
\textit{Proof.} It suffices to work with $\tau^2=\tau^2(r)$.
Differentiating $\tau^2$ we obtain %
$$ %
\frac{d\tau^2}{dr} = \frac{2r}{P(1+3r^2)^2}\, p(r), %
$$ %
where $P$ is defined by (\ref{equation P and Q}) and %
$$ %
p(r) = -5+r^2-r^4+3r^6+(3-2r^2-3r^4)P. %
$$ %
Hence, it suffices to show that $p(r)$ is negative for $0 \le r
\le 1$. It is easily seen that $P$ decreases from $\sqrt2$ to $1$
as $r$ varies from $0$ to $1$. Hence, for $0\le r\le 1$, we have
$1 \le P < 3/2$. Suppose that
\begin{eqnarray*}
c_1(r)&=&-5+r^2-r^4+3r^6 + (3-2r^2-3r^4)\cdot 1,\\
c_2(r)&=&-5+r^2-r^4+3r^6 + (3-2r^2-3r^4)\cdot (3/2).
\end{eqnarray*}
The linearity of $p$ with respect to $P$ implies that %
$$ %
\min \{c_1(r),c_2(r)\} \le p(r) \le \max \{c_1(r),c_2(r)\}. %
$$ %
Using a Sturm sequence argument, it is easily seen that both
$c_1(r)$ and $c_2(r)$ are negative for $0 \le r \le 1$. Thus,
$\tau^2(r)$ decreases on $0\le r\le 1$ and the lemma follows.
\hfill $\Box$
\bl \label{Lemma beta}%
Let $\beta=\beta(r)$ be defined by (\ref{equation Thm1-D}) with
$\tau=\tau(r)$ defined by (\ref{equation Thm1-C}). Then,
$\beta$ strictly increases from $0$ to $1$ as $r$ runs from $0$ to
$1$. %
\el %
\textit{Proof.} %
It suffices to show that $\beta^2$ is an increasing function of
$r$, which maps $[0,1]$ onto $[0,1]$. We obtain, after some
algebra, %
$$ %
\beta^2 = \frac{16 r^2 (2 r^4 + r^2 - 1) + 16 r^2 (1 + r^2) P}
{(1+r^2)^4 (1 + 2 r^2 - r^6) + (1+r^2)^4 (r^4+r^2)P}, %
$$ %
where $P$ is defined by (\ref{equation P and Q}). Differentiating
$\beta^2$, we find %
$$ %
\frac{d\beta^2}{dr} = \frac{32 r (1-r^2)^2 (1+r^2)^5 }
{P((1+r^2)^4 (1 + 2 r^2 - r^6) + (1+r^2)^4 (r^4+r^2)P)^2}\,
p(r), %
$$ %
where %
$$ %
p(r) = -4 r^6 - r^4 - 5 r^2 + 2 + (4 r^4 + 5 r^2-1)P. %
$$ %
Hence,
it suffices to show that $p(r)$ is non-negative for $0 \le r \le 1$.
Now using a Sturm sequence argument, one can finish the proof
as in the previous lemma.
\hfill $\Box$ %
\medskip
Since $r=r(h)$ is monotonic on $0 0$ on $(0,r_1)$ and $a'(r) <
0$ on $(r_1,1)$.
Using the linearity of the terms
$c_0 + c_1\,P$ and $d_0 + d_1\,P$ in $P$ and the estimates on $P$ given in
Lemma \ref{Lemma P}, one can give a Sturm sequence argument to show that
$c_0 + c_1\,P > 0$ and $d_0 + d_1\,P > 0$ on the interval $(0,r_0)$ and that
$c_0 + c_1\,P < 0$ and $d_0 + d_1\,P < 0$ on the interval $(r_2,1)$. Hence,
$a'(r) > 0$ on $(0,r_0)$ and $a'(r) < 0$ on $(r_2,1)$.
We define $n(r) = (c_0 + c_1\,P)+(d_0 + d_1\,P)G(r)$. Differentiating $n(r)$ with respect to $r$
we obtain a representation
$$n'(r) = 2r\tau^2\frac{({\tilde c}_0 + {\tilde c}_1\,P)+({\tilde d}_0 + {\tilde d}_1\,P)G(r)}{D_2(r)}$$
where the function $D_2$ is non-negative on $(0,1)$, $\tau$ is
defined by (\ref{equation Thm1-C}) and ${\tilde c}_0, \ {\tilde
c}_1, \ {\tilde d}_0$ and ${\tilde d}_1$ are polynomials in $r$
with rational coefficients.
Using the linearity of terms
${\tilde c}_0 + {\tilde c}_1\,P$ and ${\tilde d}_0 + {\tilde d}_1\,P$ in $P$ and
the estimates on $P$ given in
Lemma \ref{Lemma P}, one can give a Sturm sequence argument to show that
${\tilde c}_0 + {\tilde c}_1\,P < 0$ and ${\tilde d}_0 + {\tilde d}_1\,P < 0$ on the
interval $(r_0,r_2)$ and, hence, that $n(r)$ is strictly decreasing on the interval $(r_0,r_2)$ and
changes sign exactly once. Consequently, $a'(r)$ changes sign exactly once on $(r_0,r_2)$.
The value $r_1$ is the unique solution of $n(r) = 0$, which lies in the interval $(r_0,r_2)$.
\hfill $\Box$ %
\section{Some remarks and problems} %
\setcounter{equation}{0} %
\textbf{(a) Omitted area problem.} The following problem proposed
by A.~W.~Goodman \cite{G} can be considered as a prototype of all
omitted area problems with geometrical constraints:
%Our original interest to problems with some knowledge on a certain portion of a
%compact set came from the following problem proposed by A.~W.~Goodman:
Find $A:=\inf_{f\in S} \,\{ {\rm{Area}}\,(f(\mathbb{D})\cap
\mathbb{D})\}$ over the standard class $S$ of univalent functions
$f$ in $\mathbb{D}$ with $f(0)=0$, $f'(0)=1$.
To our knowledge, this problem remains open although many
important properties of extremal functions have been proved since
1949. Here we summarize some of them. If $f\in S$, $f$ is extremal for
$A$ and $f(1)=\infty$, then $D=f(\mathbb{D})$ is circularly
symmetric w.r.t. $\mathbb{R}_0$ %(up to rotation about the origin)
and there exist $\theta_1,\theta_2$, and $\beta$ such that
$0<\theta_1<\theta_2<\pi$, $0<\beta<1$,
and $f$ satisfies the following boundary conditions: %
\begin{enumerate} %
\item[(a)] $\Im\,f(e^{i\theta})=0$ for $0<|\theta|\le \theta_1$; %
\item[(b)] $|f(e^{i\theta})|=1$ for $\theta_1<|\theta|<\theta_2$; %
\item[(c)] $|f'(e^{i\theta})|=\beta$ for
$\theta_2<\theta<2\pi-\theta_2$; %
\item[(d)] $f'$ has a non-zero continuous extension to
$\mathbb{D}\cup \{e^{i\theta}:\,\theta_1<\theta<2\pi-\theta_1\}$
which is H\"{o}lder-continuous with exponent $1/2$; %
\item[(e)] $|f'(e^{i\theta})|$ strictly decreases in
$\theta_1<\theta<\theta_2$; %
\item[(f)] there is a $\theta_0$, $0<\theta_0<\theta_1$ such that
$|f'(e^{i\theta})|$ strictly decreases from $+\infty$ to $\beta_1$, where
$\beta_1>\beta$, and strictly increases from $\beta_1$ to $+\infty$
in $0<\theta<\theta_0$ and $\theta_0<\theta<\theta_1$,
respectively. %
\end{enumerate} %
Observations (a) and (b) were made by Barnard and Suffridge, see
\cite[p. 536]{BrC}. Condition (d) was proved by J.~Lewis \cite{L}
who also proved that (c) holds true for all $\theta$ except the
set $I=\{e^{i\theta}:\,\Im\,f(e^{i\theta})=0\}$ which may consists
of at most a finite number of closed arcs. The inequality
$\beta<1$ and conditions (e), (f), and (c) without the above
mentioned exception were established in \cite{BS}.
\smallskip
The conclusion of Lemma~\ref{Lemma 2.2}\textbf{(d)}
that the vertical non-free boundary is not degenerate, i.e., that
there is a strict inequality $\theta_1<\theta_2$ for
the parameters $\theta_1$ and $\theta_2$ of this iceberg-type problem is reminscient of the
conclusion in \cite{L} that there is a strict inequality $\theta_1<\theta_2$ for
the parameters $\theta_1$ and $\theta_2$ of Goodman's omitted
area problem. With minor modifications, the proof in \cite{L} that
$\theta_1<\theta_2$ for Goodman's omitted area problem could have been modified to
prove Lemma~\ref{Lemma 2.2}\textbf{(d)}. In this paper, we have given an independent
proof of Lemma~\ref{Lemma 2.2}\textbf{(d)} and we mention here that, alternatively,
with minor modifications the proof of Lemma~\ref{Lemma 2.2}\textbf{(d)} could be used
to show that $\theta_1<\theta_2$ for Goodman's omitted area problem as well.
Approximations to the exact value of $A$ have been given in \cite{BP,BT} by
different numerical methods. In particular, \cite{BT} suggests that
$A = 0.2385813284\pi$, where all explicitly shown digits are exact.
\medskip
\textbf{(b) Width of the invisible part of the iceberg.} The
method of this paper can be also applied to find the extremal
function for Problem~(\ref{equation maximals})\textbf{(b)} if one
can show a priori that the free boundary of the extremal is smooth enough. One
difference compared to Problem~(\ref{equation
maximals})\textbf{(a)} is that the extremal configurations now
\emph{do not} possess circular symmetry although they still
possess Steiner symmetry. In view of this lack of symmetry, we cannot
apply the local variations developed in
Section~\ref{Section 2} since the boundary may be non-rectifiable.
Perhaps, the necessary smoothness can be achieved by applying a
more powerful technique such as that of J.~Lewis \cite{L}
mentioned in the Introduction.
\medskip
\textbf{(c) Safe distance from the iceberg.} The situation with
Problem~(\ref{equation maximals})\textbf{(c)} differs from the
other two cases. To explain this, we start with the limiting case
when the whole iceberg is observable, i.e. when $\Area (E_+)=\pi$.
Then, of course, $E$ coincides with the disk $\{w:\,|w-(1+i)|\le
1\}$ up to translation along the real axis.
This disk has a contact point with the surface of interface at
$z=1$ and a contact point with the front line, which coincides
with the imaginary axis, at $z=i$. These two contact points
represent the non-free boundary in this limiting case. It is
reasonable to expect that for icebergs with visible area slightly
less than $\pi$, the extremal configurations will have two
disjoint segments, vertical and horizontal, as their non-free
boundary. If so, then transplanting the problem into the auxiliary
$s$-plane as in Section~\ref{Section 3}, we have to deal with the
omitted area problem for functions defined in a doubly-connected
domain. To our knowledge, there are no known solutions of
problems of this kind.
\medskip
\textbf{(d) Convex icebergs.} Let ${\mathcal{F}}^c$ denote the
collection of all convex compact sets $E$ in ${\mathcal{F}}$. It
will be interesting to study problems (\ref{equation maximals})
for the class ${\mathcal{F}}^c$. Since there are more available
methods for convex sets and functions, there is a chance that
known techniques may give complete solutions to all three
problems.
\begin{thebibliography}{6.8in}
\bibitem{AShS1}
D.~Aharonov, H.~S.~Shapiro, and A.~Yu.~Solynin,
\textit{A minimal area problem in conformal mapping}. J. Analyse Math.
\textbf{78} (1999), 157--176.
\bibitem{AShS2}
D.~Aharonov, H.~S.~Shapiro, and A.~Yu.~Solynin,
\textit{Minimal area problems for functions with integral representation}. J. Analyse Math.
\textbf{98} (2006), 83--111.
\bibitem{BT} L. Banjai, L. Trefethen,
\textit{Numerical solution of the omitted area problem of univalent function theory.} Comp. Methods Func. Theory
\textbf{1} (2001), no. 1, 259--273.
\bibitem{BL} R. W. Barnard, J. L. Lewis,
\textit{On the omitted area problem.} Michigan Math. J.
\textbf{34} (1987), 13--22.
\bibitem{BP} R. W. Barnard, K. Pearce,
\textit{Rounding Corners of Gearlike Domains and the Omitted Area Problem.} J. Comp. Appl. Math.
\textbf{14} (1986), 217--226;
\textit{Numerical Conformal Mapping.} North Holland, 1986.
\bibitem{BPS1} R. W. Barnard, K. Pearce, and A. Yu. Solynin,
\textit{An isoperimetric inequality for logarithmic capacity.} Annales Academi\ae \ Scientiarum Fennic\ae. Mathematica
\textbf{27} (2002), 419--436.
\bibitem{BRS1} R. W. Barnard, C. Richardson, and A. Yu. Solynin,
\textit{Concentration of area in half-planes.} Proc. Amer. Math. Soc.
\textbf{133} (2005), no. 7, 2091--2099.
\bibitem{BRS2} R. W. Barnard, C. Richardson, and A. Yu. Solynin,
\textit{A minimal area problem for nonvanishing functions.} Algebra i Analiz
\textbf{18} (2006), no. 1, 35-54;
English translation in: St. Petersburg Math. J.
\textbf{18} (2007), no. 1, 21--36.
\bibitem{BS} R. W. Barnard and A. Yu. Solynin,
\textit{Local variations and minimal area problem for Carath\'{e}odory functions.} Indiana U. Math. J.
\textbf{53} (2004), no. 1, 135--167.
\bibitem{BrC} D. Brannan and J.~Clunie,
\textit{Aspects of Contemporary Complex Analysis.} Academic Press, New York, 1980.
\bibitem{D} V. N. Dubinin,
\textit{Symmetrization in geometric theory of functions of a complex variable.} Uspehi Mat. Nauk
\textbf{49} (1994), 3-76 (in Russian);
English translation in: Russian Math. Surveys
\textbf{49}: 1 (1994), 1--79.
\bibitem{Du} P. Duren,
\textit{Univalent Functions.} Springer-Verlag, 1992.
\bibitem{G} A. Goodman,
\textit{Note on regions omitted by univalent functions.} Bull. Amer. Math. Soc.
\textbf{55} (1949), 363--369.
\bibitem{H} W. K. Hayman,
\textit{Multivalent Functions.} Second Edition. Cambridge Tracts in Mathematics, 110. Cambridge Univ. Press, Cambridge, 1994.
\bibitem{J} N. Jacobson,
\textit{Basic Algebra. I.} Second Edition. W. H. Freeman and Company, New York, 1985. % MR0780184 (86d:00001)
\bibitem{L} J. L. Lewis,
\textit{On the minimal area problem.} Indiana Univ. Math. J.
\textbf{34} (1985), 631--661.
\bibitem{P} Ch. Pommerenke,
\textit{Boundary Behaviour of Conformal Maps.} Springer-Verlag, 1992.
ath. J. \textbf{8} (1997), 1015--1038.
%
\bibitem{S} A. Yu. Solynin,
\textit{A Functional inequalities via polarization.} Algebra i Analiz
\textbf{8} (1996), 145-185;
English translation in: St. Petersburg Math. J.
\textbf{8} (1997), 1015--1038.
\end{thebibliography}
\end{document}