### Abstract

We consider the complex plane ℂ as a space filled by two different media, separated by the real axis ℝ. For a planar body (connected compact set) E in C, the sets E_{+}=E∩{z:>0} and E_{-}=E\{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{:z∈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 class of sectorial starlike functions.

Original language | English |
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Pages (from-to) | 635-682 |

Number of pages | 48 |

Journal | Computational Methods and Function Theory |

Volume | 13 |

Issue number | 4 |

DOIs | |

State | Published - 2013 |

### Keywords

- Convex function
- Iceberg-type problem
- Logarithmic capacity
- Omitted area problem
- Starlike function

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## Cite this

*Computational Methods and Function Theory*,

*13*(4), 635-682. https://doi.org/10.1007/s40315-013-0042-y