Limit of star-likeness

exact radius of star-likeness, bound of star-likeness

The least upper bound $R_U$ of the radii of discs $|z|\leq r$, where $U$ is some class of functions $w=f(z)+\dotsb$ that are regular and univalent in $|z|<1$, such that the functions from $U$ on the disc $|z|<1$ map the discs $|z|\leq r$ onto star-like domains (cf. Star-like domain) about the point $w=0$. Any number $r$ in the interval $0<r<R_U$ is called a radius of star-likeness of the class $U$.

The limit of star-likeness is usually found by using the following criterion of star-likeness: A disc $|z|<r$ is mapped onto a star-like domain by $w=f(z)$ if and only if on $|z|=r$,

$$\frac{\partial\arg f(z)}{\partial\phi}=\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]\geq0,\quad z=re^{i\phi},$$

or, equivalently,

$$\left|\arg\frac{zf'(z)}{f(z)}\right|\leq\frac\pi2.$$

The limit of star-likeness $R_S$ of the class $S$ of all functions $f(z)=z+\dotsb$ that are regular and univalent in the disc $|z|<1$ is equal to $\tanh(\pi/4)=0.65\dots$.

References

[1]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11