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) |
Comments
References
[a1] | P.L. Duren, "Univalent functions" , Springer (1983) pp. Chapt. 10 |
Bieberbach-Eilenberg functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bieberbach-Eilenberg_functions&oldid=16091