TY - JOUR

T1 - Convex icebergs and sectorial starlike functions

AU - Barnard, Roger W.

AU - Lochman, Matthew H.

AU - Solynin, Alexander Yu

N1 - Funding Information:
Research of the Alexander Yu. Solynin was supported in part by NSF Grant DMS-1001882. The work of the Matthew H. Lochman constitutes part of his PhD Dissertation defended at Texas Tech University, May 2011.

PY - 2013/12

Y1 - 2013/12

N2 - 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.

AB - 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.

KW - Convex function

KW - Iceberg-type problem

KW - Logarithmic capacity

KW - Omitted area problem

KW - Starlike function

UR - http://www.scopus.com/inward/record.url?scp=84889647445&partnerID=8YFLogxK

U2 - 10.1007/s40315-013-0042-y

DO - 10.1007/s40315-013-0042-y

M3 - Article

AN - SCOPUS:84889647445

SN - 1617-9447

VL - 13

SP - 635

EP - 682

JO - Computational Methods and Function Theory

JF - Computational Methods and Function Theory

IS - 4

ER -