TY - JOUR

T1 - Convex icebergs and sectorial starlike functions.

AU - Barnard, Roger

AU - Lochman, Matthew

AU - Solynin, Alexander

PY - 2013/4

Y1 - 2013/4

N2 - We consider the complex plane $\mathbb{C}$ as a space filled by
two different media, separated by the real axis $\mathbb{R}$. For
a planar body (connected compact set) $E$ in $\mathbb{C}$, the
sets $E_+=E\cap\{z:\Im z>0\}$ and $E_-=E\backslash\{z:\Im z>0\}$
will be considered as a ``visible'' part of $E$ and an
``invisible'' part of $E$, respectively. In a recent paper of
Barnard, Pearce and Solynin, the authors discussed the problem of
estimating the maximum draft, $\min\{\Im z:z\in E_-\}$, from
characteristics of the whole body $E$ and characteristics of its
``visible'' part $E_+$. Their study was focused on the general
case of compact sets, i.e. they worked with sets $E$ which are not
necessarily convex.
In this paper, we investigate the maximal draft of $E$ as a
function of the logarithmic capacity of $E$ and the area of $E_+$
under the more natural, yet also more difficult, assumption that
$E$ is convex. As a tool in our proof, we consider extremal
problems for the related cla

AB - We consider the complex plane $\mathbb{C}$ as a space filled by
two different media, separated by the real axis $\mathbb{R}$. For
a planar body (connected compact set) $E$ in $\mathbb{C}$, the
sets $E_+=E\cap\{z:\Im z>0\}$ and $E_-=E\backslash\{z:\Im z>0\}$
will be considered as a ``visible'' part of $E$ and an
``invisible'' part of $E$, respectively. In a recent paper of
Barnard, Pearce and Solynin, the authors discussed the problem of
estimating the maximum draft, $\min\{\Im z:z\in E_-\}$, from
characteristics of the whole body $E$ and characteristics of its
``visible'' part $E_+$. Their study was focused on the general
case of compact sets, i.e. they worked with sets $E$ which are not
necessarily convex.
In this paper, we investigate the maximal draft of $E$ as a
function of the logarithmic capacity of $E$ and the area of $E_+$
under the more natural, yet also more difficult, assumption that
$E$ is convex. As a tool in our proof, we consider extremal
problems for the related cla

M3 - Article

SP - 635

EP - 682

JO - Computational Methods and Function Theory

JF - Computational Methods and Function Theory

ER -