Subordination families and extreme points. (English) Zbl 0651.30019

Let F be a univalent in the unit disk \(\Delta\), let s(F) denote the family of functions in \(\Delta\) that are subordinate to F, and let \(B_ 0=s(id).\)
The authors prove that if \(f=F\circ \phi\) is an extreme point of s(F) then \(\Phi\) must be extreme in \(B_ 0\), therefore solving a conjecture made by the first author in [Proc. Am. Math. Soc. 87, 439-443 (1983; Zbl 0519.30023)]. If \(\lambda\) (w,\(\partial F(\Delta))\) denotes the distance between w and \(\partial F(\Delta)\) it is also shown that for each extreme point \(\phi\) of \(B_ 0\) the relation \[ \int^{2\pi}_{0}\log \lambda (F(e^{i\theta}\phi (e^{it})),\quad \partial F(\Delta))dt=-\infty \] holds for almost all \(\theta\).
Reviewer: W.Koepf


30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30C55 General theory of univalent and multivalent functions of one complex variable


Zbl 0519.30023
Full Text: DOI


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