On some properties of schlicht functions

One considers the classes $S_\beta^*(\alpha)$, $S_\beta(\gamma)$ and $S$ of functions $f(z)=z+\dots$, which are respectively $\alpha$-starlike of orderbeta, $\gamma$-spirallike of order $\beta$, and regular schlicht in $|z|<1$. It is proved that for $\alpha\ge0$, $0<\beta<1$ from $f(z)\in S^*_\beta(\alpha)$ follows $f(z)\in S_\beta^*(0)$; this generalizes appropriate results of [1–5]. A converse result is also obtained. For certain $\alpha$ and $\beta$ the exact value of the radius of $\alpha$-starlikeness of orderbeta for the class $S$ is given. An equation is found, whose unique root gives the radius $\gamma$-spirallikeness of order $\beta$ for the class $S$.