Univalency radius

radius of univalence

The radius $\rho(M)$ of the largest disc $|z|<\rho$ in which all functions of the form

$$f(z)=z+a_2z^2+\dotsb+a_nz^n+\dotsb$$

belonging to the family of functions that are regular in the disc $|z|<1$ satisfying $|f(z)|\leq M$ for $|z|<1$ are univalent. It turns out that

$$\rho(M)=M-\sqrt{M^2-1},\quad M\geq1,$$

and the function

$$Mz\frac{1-Mz}{M-z}$$

is univalent in the disc $|z|<\rho(M)$, but not in any larger disc (with centre at the origin). For functions regular in the disc $|z|<R$ and such that $f(0)=0$, $f'(0)=c$, $c\neq0$, and $|f(z)|\leq M$, the radius of univalence $\rho^*(M)$ is defined similarly, and its value can be easily obtained from $\rho(M)$.

Comments

Cf. also Univalency conditions; Univalent function.

References

[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11
[a2]	A.W. Goodman, "Univalent functions" , 2 , Mariner (1983)
[a3]	E. Landau, "Der Picard–Schottkysche Satz und die Blochse Konstante" Sitzungsber. Akad. Wiss. Berlin Phys. Math. Kl. (1925) pp. 467–474