Convex hulls and extreme points of families of symmetric univalent functions

Bull. Korean Math. Soc. 1996 Vol. 33, No. 1, 1-16

J. S. Hwang Academia Sinica

Abstract : Let $Q_k$ be the set of all functions $f$ analytic in $D(\vert z\vert <1)$ such
that $f(0)=0, f'(0)=1$, and $f(z)$ is in the $j$-th sector whenever $z$ is in
the $j$-the sector of $D$, for $j=1, 2, \cdots, k$. Let $U_k$ be the set of
extreme points of $Q_k$ which are univalent in $D$. We prove that if $k$ is
odd, then $U_k$ contains exactly two elements, i.e. $z/(1+z^{2k})^{1/k}$ and
$z/(1-z^k)^{2/k}$. If $k$ is even, then $f\in U_k$ if and only if
$f(z)=z/[(1-e^{ik\theta} z^k) (1-e^{ik\theta} z^k)]^{1/k}$, for some $0\le
\theta < 2\pi$. Let $U_k^*$ be the closed convex hull of $U_k$. Then for any
$f(z)=\sum_{n=0}^\infty a_{nk+1} z^{nk+1}$ in $U_k^*$, we have
$$\vert a_{nk+1}\vert \le 2(2+k) (2+2k)\cdots (2+(n-1)k)/(k^n n!) \le 2/k\qquad
\text{for}\ \ n=1, 2, \cdots .$$

Keywords : Convex hull, extreme point, symmetric and univalent function, coefficient estimate, and cluster set