On nonvanishing univalent functions with real coefficients (Q1079978)

Let \(S_ 0(R)\) be the class of all functions analytic and univalent in the unit disc D that satisfy the conditions \[ (i)\quad f(z)=1+\sum^{\infty}_{k=1}a_ kz^ k,\quad (ii)\quad a_ k=\bar a_ k,\quad (iii)\quad 0\not\in f({\mathbb{D}}). \] The author shows that (i) every extreme point of \(S_ 0(R)\cup \{1\}\) has the form \[ (1+z)^ 2[(1-yz)(1-\bar yz)]^{-1}\quad or\quad (1-z)^ 2[(1-yz)(1-\bar yz)]^{-1},\quad y\in \partial {\mathbb{D}}\setminus \{1\}. \] (ii) Every support point of \(S_ 0({\mathbb{R}})\) has the form \[ 1+kz[(1-yz)(1- yz)]^{-1}\text{ for some }y\in \partial {\mathbb{D}} \] and \(k\in [-2(1-Re y),2(1+Re y)]\), \(k\neq 0\). The main tool used here is a result of \textit{L. Brickman} [Bull. Am. Math. Soc. 76, 372-374 (1970; Zbl 0189.088)].