View source for 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|Univalent function]]) in  $  R $
is denoted by  $  \widetilde{R}  $. 
The functions in  $  R $
were named after L. Bieberbach [[#References|[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 [[#References|[2]]], who proved that the inequality (2) is valid for the whole class  $  R $. 
It was shown by W. Rogosinski [[#References|[3]]] that every function in  $  R $
is subordinate (cf. [[Subordination principle|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|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|quadratic differential]] in the  $  w $-
plane with a certain symmetry in the distribution of the zeros and poles [[#References|[4]]], [[#References|[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 [[#References|[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 [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bieberbach,   "Ueber einige Extremalprobleme im Gebiete der konformen Abbildung"  ''Math. Ann.'' , '''77'''  (1916)  pp. 153–172</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Eilenberg,   "Sur quelques propriétés topologiques de la surface de sphère"  ''Fund. Math.'' , '''25'''  (1935)  pp. 267–272</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Rogosinski,   "On a theorem of Bieberbach–Eilenberg"  ''J. London Math. Soc. (1)'' , '''14'''  (1939)  pp. 4–11</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.A. Jenkins,   "Univalent functions and conformal mappings" , Springer  (1958)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.A. Jenkins,   "On Bieberbach–Eilenberg functions III"  ''Trans. Amer. Math. Soc.'' , '''119''' :  2  (1965)  pp. 195–215</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N.A. Lebedev,   "The area principle in the theory of univalent functions" , Moscow  (1975)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Duren,   "Univalent functions" , Springer  (1983)  pp. Chapt. 10</TD></TR></table>