# Bieberbach-Eilenberg functions

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in the disc $| z | < 1$

The class $R$ of functions $f(z)$, regular in the disc $| z | < 1$, which have an expansion of the form

$$\tag{1 } f(z) = c _ {1} z + \dots + c _ {n} z ^ {n} + \dots$$

and which satisfy the condition

$$f(z _ {1} ) f (z _ {2} ) \neq 1 ,\ \ | z _ {1} | < 1,\ | z _ {2} | < 1.$$

This class of functions is a natural extension of the class $B$ of functions $f(z)$, regular in the disc $| z | < 1$, with an expansion (1) and such that $| f(z) | < 1$ for $| z | < 1$. The class of univalent functions (cf. Univalent function) in $R$ is denoted by $\widetilde{R}$. The functions in $R$ were named after L. Bieberbach [1], who showed that for $f(z) \in \widetilde{R}$ the inequality

$$\tag{2 } | c _ {1} | \leq 1$$

is valid, while equality holds only for the function $f(z) = e ^ {i \theta } z$, where $\theta$ is real, and after S. Eilenberg [2], who proved that the inequality (2) is valid for the whole class $R$. It was shown by W. Rogosinski [3] that every function in $R$ is subordinate (cf. Subordination principle) to some function in $\widetilde{R}$. Inequality (2) yields the following sharp inequality for $f(z) \in R$:

$$\tag{3 } | f ^ { \prime } (z) | \leq \ \frac{| 1 - f ^ {2} (z) | }{1- | z | ^ {2} } ,\ \ | z | < 1.$$

The following bound on the modulus of a function in $R$ has been obtained: If $f(z) \in R$, then

$$\tag{4 } | f(z) | \leq \frac{r}{(1-r ^ {2} ) ^ {1/2} } ,\ \ | z | = r ,\ 0 < r < 1,$$

and (4) becomes an equality only for the functions $\pm f(ze ^ {i \theta } ; r)$, where $\theta$ is real and

$$f (z; r) = \ \frac{(1 - r ^ {2} ) ^ {1/2} z }{1 + irz } .$$

The method of the extremal metric (cf. Extremal metric, method of the) provided the solution of the problem of the maximum and minimum of $| f(z) |$ in the class $\widetilde{R} (c)$ of functions in $\widetilde{R}$ with a fixed value $| c _ {1} | = c$, $0 < c \leq 1$, in the expansion (1): For $f(z) \in \widetilde{R} (c)$, $0 < c < 1$, the following sharp inequalities are valid:

$$\tag{5 } \mathop{\rm Im} H (ir; r, c) \leq \ | f (re ^ {i \theta } ) | \leq \ \mathop{\rm Im} F (ir; r, c).$$

Here the functions $w = H(z; r, c)$ and $w = F(z; r, c)$ map the disc $| z | < 1$ onto domains which are symmetric with respect to the imaginary axis of the $w$- plane, and the boundaries of which belong to the union of the closures of certain trajectories or orthogonal trajectories of a quadratic differential in the $w$- plane with a certain symmetry in the distribution of the zeros and poles [4], [5]. Certain optimal results for functions in $\widetilde{R} (c)$ were obtained by the simultaneous use of the method of the extremal metric and the symmetrization method [4].

Many results obtained for the functions in the classes $\widetilde{R}$ and $R$ are consequences of corresponding results for systems of functions mapping the disc $| z | < 1$ onto disjoint domains [6]. The analogue of $R$ for a finitely-connected domain $G$ without isolated boundary points and not containing the point $z = \infty$, is the class $R _ {a} (G)$, $a \in G$, of functions $f(z)$ regular in $G$ and satisfying the conditions $f(a) = 0$, $f(z _ {1} )f(z _ {2} ) \neq 1$, where $z _ {1} , z _ {2}$ are arbitrary points in $G$. The class $R _ {a} (G)$ extends the class $B _ {a} (G)$ of functions $f(z)$, regular in $G$ and such that $f(a) = 0$, $| f(z) | < 1$ in $G$. The following sharp estimate is an extension of the result of Bieberbach–Eilenberg and of inequality (3) to functions of class $R _ {a} (G)$: If $f(z) \in R _ {a} (G)$, then

$$| f ^ { \prime } (z) | \leq \ | 1 - f ^ { 2 } (z) | \ F ^ { \prime } (z, z),\ \ z \in G.$$

where $F(z, b), b \in G$, is that function in $B _ {b} (G)$ for which $F ^ { \prime } (b, b) = \max | f ^ { \prime } (b) |$ in this class.

#### References

 [1] L. Bieberbach, "Ueber einige Extremalprobleme im Gebiete der konformen Abbildung" Math. Ann. , 77 (1916) pp. 153–172 [2] S. Eilenberg, "Sur quelques propriétés topologiques de la surface de sphère" Fund. Math. , 25 (1935) pp. 267–272 [3] W. Rogosinski, "On a theorem of Bieberbach–Eilenberg" J. London Math. Soc. (1) , 14 (1939) pp. 4–11 [4] J.A. Jenkins, "Univalent functions and conformal mappings" , Springer (1958) [5] J.A. Jenkins, "On Bieberbach–Eilenberg functions III" Trans. Amer. Math. Soc. , 119 : 2 (1965) pp. 195–215 [6] N.A. Lebedev, "The area principle in the theory of univalent functions" , Moscow (1975) (In Russian)