On an extreme point conjecture for concave functions
Let \(1< A\leq 2\) be a fixed number, let \(\mathbb{D}= \{z: |z|< 1\}\) denote the open unit disk, \(\varphi(z): \mathbb{D}\to\overline{\mathbb{D}}\) be a holomorphic function, and define \[ \mathrm{CO}(A):= \Biggl\{f:{f''(z)\over f'(z)}= {A+1\over z-1}+ {(A-1)\varphi(z)\over 1+z\varphi(z)}\Biggr\}. \] Functions in \(\mathrm{CO}(A)\) map \(\mathbb{D}\) onto domains \(f(\mathbb{D})\) that have the opening angle at infinity of measure less than \(\pi A\) and such that the sets \(C\setminus f(\mathbb{D})\) are convex. It was conjectured by \textit{B. Bhowmik} and the authors [Indian J. Math. 50, No. 2, 339--349 (2008; Zbl 1161.30006)] that functions in \(\mathrm{CO}(A)\) that map \(\mathbb{D}\) onto the exterior of a convex unbounded polygon are among the extreme points of the closed convex hull of \(\mathrm{CO}(A)\). The conjecture remains open. The present paper bring a justification of its weak form.
scientific article; zbMATH DE number 6108758