Bieberbach-Eilenberg functions

in the disc

The class of functions , regular in the disc , which have an expansion of the form

(1)

and which satisfy the condition

This class of functions is a natural extension of the class of functions , regular in the disc , with an expansion (1) and such that for . The class of univalent functions (cf. Univalent function) in is denoted by . The functions in were named after L. Bieberbach [1], who showed that for the inequality

(2)

is valid, while equality holds only for the function , where is real, and after S. Eilenberg [2], who proved that the inequality (2) is valid for the whole class . It was shown by W. Rogosinski [3] that every function in is subordinate (cf. Subordination principle) to some function in . Inequality (2) yields the following sharp inequality for :

(3)

The following bound on the modulus of a function in has been obtained: If , then

(4)

and (4) becomes an equality only for the functions , where is real and

The method of the extremal metric (cf. Extremal metric, method of the) provided the solution of the problem of the maximum and minimum of in the class of functions in with a fixed value , , in the expansion (1): For , , the following sharp inequalities are valid:

(5)

Here the functions and map the disc onto domains which are symmetric with respect to the imaginary axis of the -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 -plane with a certain symmetry in the distribution of the zeros and poles [4], [5]. Certain optimal results for functions in 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 and are consequences of corresponding results for systems of functions mapping the disc onto disjoint domains [6]. The analogue of for a finitely-connected domain without isolated boundary points and not containing the point , is the class , , of functions regular in and satisfying the conditions , , where are arbitrary points in . The class extends the class of functions , regular in and such that , in . The following sharp estimate is an extension of the result of Bieberbach–Eilenberg and of inequality (3) to functions of class : If , then

where , is that function in for which 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