Series of univalent functions

B. N. Rakhmanov

Matematičeskie zametki, Tome 23 (1978) no. 4, pp. 593-600

Résumé

We are considering a class $S$ of functions $F(z)$, $F(0)=0$, $F'(0)=1$ that are univalent and regular in the circle $|z|=1$, and its subclasses $S_h^*$ and $K$ of starlike functions of order $h$ and of convex functions respectively. Among others, we establish the following results: If $F(z)\in S$ and $0<\alpha<1$, then $$ \biggl|\frac{\alpha zF''(\alpha z)}{F'(\alpha z)}-\frac{zF''(z)}{F'(z)}+\frac2{1-r^2}-\frac2{1-\alpha^2r^2}\biggr|\le\frac{4r}{1-r^2}-\frac{4\alpha r}{1-\alpha^2r^2},\quad|z|=r $$ If $F(z)\in S$ ($0<\alpha<1$) and $$ 1+\operatorname{Re}z_1F''(z_1)/F'(z_1)=\operatorname{Re}\alpha z_1F''(\alpha z_1)/F'(\alpha z_1)\quad(2-\sqrt3<|z_1|=r<1), $$ then we obtain the domain of values of the point $\alpha z_1$.