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 -