New support points of $\mathcal {S}$ and extreme points of $\mathcal {HS}$

Abstract:

Let $\mathcal {S}$ be the usual class of univalent analytic functions $f$ on $\{ z\left | {|z|} \right . < 1\}$ normalized by $f(z) = z + {a_2}{z^2} + \cdots$. We prove that the functions \[ {f_{x,y}}(z) = \frac {{z - \tfrac {1} {2}(x + y){z^2}}} {{{{(1 - yz)}^2}}},\quad \left | x \right | = \left | y \right | = 1,x \ne y,\] which are support points of $\mathcal {C}$, the subclass of $\mathcal {S}$ of close-to-convex functions, and extreme points of $\mathcal {H}\mathcal {C}$, are support points of $\mathcal {S}$ and extreme points of $\mathcal {H}\mathcal {S}$ whenever $0 < \left | {\arg ( - x/y)} \right | \leqslant \pi /4$. We observe that the known bound of $\pi /4$ for the acute angle between the omitted arc of a support point of $\mathcal {S}$ and the radius vector is achieved by the functions ${f_{x,y}}$ with $\left | {\arg ( - x/y)} \right | = \pi /4$.