Bieberbach-Eilenberg functions

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)

References

[a1]	P.L. Duren, "Univalent functions" , Springer (1983) pp. Chapt. 10