İstanbul Analysis Seminars

Cross ratio problem on some subclasses of univalent functions
Yusuf Avcı
Nişantaşı University, Turkey
Özet : Let $D$ be a simply connected domain in the closed complex plane $\mathbb{C}$. For given distinct poins $z_k\, (k=1,...,4)$ in $D$, let $(z_1,z_2,z_3, z_4)$ denote their cross ratio. It is well-known that if $w=h(z)$ is a M\" obius transformation and $w_{k}=h(z_{k})$, then $(z_1, z_2, z_3, z_4)=(w_1, w_2, w_3, w_4)$ but it may not be so if $h$ is an arbitrary univalent function in $D$. If $w=f(z)$ is univalent in $D$, then the quotient $(w_1, w_2, w_3, w_4)/(z_1, z_2, z_3, z_4)$, denoted by $Q(f)$, determines a {\bf\emph{measure of deviation}} of $f$ from a M\" obius transformation. So there arises the problem of finding the variability region of $Q(f)$ when $f$ belongs to a class of univalent functions which is compact in the space $H(D)$ of holomorphic functions in $D$. In this talk we will point out a couple of results in this direction and give a concrete result for a small class ``univalent second degree polynomials.''
  Tarih : 15.04.2016
  Saat : 15:40
  Yer : Sabancı University, Karaköy Communication Center, Bankalar Caddesi 2, Karaköy
  Dil : English
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